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For any scalar c, [u]*(c[v]) = c([u]*[v]).

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If A is an mxn matrix, then the range of the transformation [x] -> A[x] is in Rm.

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If A is an nxn diagonalizable matrix, then each vector in Rn can be written as a linear combination of eigenvectors of A.

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If A is invertible, then (detA)(det(A^-1)) = 1

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If A is invertible, then det(A^-1) = detA

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If A is mxn and rankA = m, then the linear transformation [x] -> A[x] is one-to-one.

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If A is nxn and detA = 2, then det(A^3) = 6

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If BC = BD, then C = D.

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If L is a line through [0] and if y-hat is the orthogonal projection of [y] onto L, then ||y-hat|| gives the distance from [y] to L.

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If Pb is the change-of-coordinates matrix, then [[x]]b=Pb[x], for [x] in V.

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If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.

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If T: Rn -> Rm is alinear transformation and if [c] is in Rm, then a uniqueness question is "Is [c] in the range of T?"

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If V=R2, B={[b]1, [b]2}, and C={[c]1, [c]2}, then row reduction of [ [c]1 [c]2 [b]1 [b]2 ] to [ I P ] produces a matrix P that satisfies [[x]]b = P[[x]]c for all [x] in V.

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If V=Rn and C is the standard basis for V, then P(C<-B) is the same as the change-of-coordinates matrix Pb introduced in Section 4.4.

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If [x] is a nontrivial solution of A[x] = [0], then every entry in [x] is nonzero.

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If [y] = [z]1 + [z]2, where [z]1 is in a subspace W and [z]2 is in W perp, then [z]1 must be the orthogonal projection of [y] onto W.

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If [y] is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.

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If [y] is in a subspace W, then the orthogonal projection of [y] onto W is [y] itself.

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If [z] is orthogonal to [u]1 and to [u]2 and if W = Span {[u]1, [u]2}, then [z] must be in W perp.

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If a 5x5 matrix A has fewer than 5 distinct eigenvalues, then A is not diagonalizable.

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If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.

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The dimension of the null space of A is the number of columns of A that are not pivot columns.

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The dimension of the vector space P4 is 4.

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The dimensions of the row space and the column space of A are the same, even if A is not square.

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The echelon form of a matrix is unique.

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The effect of adding [p] to a vector is to move the vector in a direction parallel to [p].

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The eigenvalues of an upper triangular matrix A are exactly the nonzero entries on the diagonal of A.

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The equation A[x] = [0] gives an explicit description of its solution set.

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The general least-squares problem is to find an [x] that makes A[x] as close as possible to [b].

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The homogeneous equation A[x] = [0] has the trivial solution if and only if the equation has at least one free variable.

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The kernel of a linear transformation is a vector space.

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The least-squares solution of A[x] = [b] is the point in the column space of A closest to [b].

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The null space of A is the solution set of the equation A[x] = [0].

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The null space of an mxn matrix is in Rm.

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The number of pivot columns of a matrix equals the dimension of its column space.

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The number of variables in the equation A[x]=[0] equals the dimension of NulA.

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The only three-dimensional subspace of R3 is R3 itself.

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The orthogonal projection of [y] onto [v] is the same as the orthogonal projection of [y] onto c[v] whenever c does not equal 0.

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The orthogonal projection y-hat of [y] onto a subspace W can sometimes depend on the orthogonal basis for W used to compute y-hat.

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The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.

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The points in the plane corresponding to (-2, 5) and (-5, 2) lie on a line through the origin.

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The range of a linear transformation is a vector space.

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The solution set of A[x] = [b] is the set of all vectors of the form [w] = [p] + [v]h, where [v]h is any solution of the equation A[x] = [0].

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The solution set of a linear system involving variables x1, ... , xn is a list of numbers (s1, ... , sn) that makes each equation in the system a true statement when the values s1, ... , sn are substituted for x1, ... , xn, respectively.

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The solution set of a linear system whose augmented matrix is [ [a]1 [a]2 [a]3 [b] ] is the same as the solution set of A[x] = [b], if A = [ [a]1 [a]2 [a]3 ].

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Two vectors are linearly dependent if and only if they lie on a line through the origin.

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When [u] and [v] are nonzero vectors, Span {[u], [v]} contains the line through [u] and the origin.

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When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.

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Whenever a system has free variables, the solution set contains many solutions.

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[u]*[v] - [v]*[u] = 0.

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[v]*[v] = ||[v]||^2.

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Two eigenvectors corresponding to the same eigenvalue are always linearly dependent.

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Two fundamental questions about a linear system involve existence and uniqueness.

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Two linear systems are equivalent if they have the same solution set.

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Two matrices are row equivalent if they have the same number of rows.

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A (square) matrix A is invertible if and only if there is a coordinate system in which the transformation [x] -> A[x] is represented by a diagonal matrix.

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A 5x6 matrix has 6 rows.

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A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

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A basis is a linearly independent set that is as large as possible.

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A basis is a spanning set that is as large as possible.

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A change-of-coordinates matrix is always invertible.

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A general solution of a system is an explicit description of all solutions of the set.

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A homogeneous equation is always consistent.

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A least squares solution of A[x] = [b] is a vector x-hat such that ||[b] - A[x]|| is less than or equal to ||[b] - A(x-hat)|| for all [x] in Rn.

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A least-squares solution of A[x] = [b] is a list of weights that, when applied to the columns of A, produces the orthogonal projection of [b] onto ColA.

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A least-squares solution of A[x] = [b] is a vector x-hat that satisfies A(x-hat) = b-hat, where b-hat is the orthogonal projection of [b] onto ColA.

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A linear transformation T: Rn -> Rm is completely determined by its effect on the columns of the nxn identity matrix.

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A linear transformation is a special type of function.

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A linear transformation preserves the operations of vector addition and scalar multiplication.

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A linearly independent set in a subspace H is a basis for H.

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A mapping T: Rn -> Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm.

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A mapping T: Rn -> Rm is onto Rm if every vector [x] in Rn maps onto some vector in Rm.

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A matrix with orthonormal columns is an orthogonal matrix.

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A nonzero vector cannot correspond to two different eigenvalues of A.

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A null space is a vector space.

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A plane in R3 is a two-dimensional subspace of R3.

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A plane in R3 is a two-dimensional subspace.

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A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order.

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A row replacement operation does not affect the determinant of a matrix

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A single vector by itself is linearly dependent.

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A subset H of a vector space V is a subspace of V if the zero vector is in H.

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A transformation T is linear if and only if T(c1[v]1 + c2[v]2) = c1T([v]1) + c2T([v]2) for all [v]1 and [v]2 in the domain of T and for all scalars c1 and c2.

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A vector is any element of a vector space.

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A vector space is also a subspace.

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A vector space is infinite-dimensional if it is spanned by an infinite set.

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A^t + B^t = (A+B)^t

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An elementary matrix must be square.

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An elementary nxn matrix has either n or n+1 nonzero entries.

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An example of a linear combination of vectors [v]1 and [v]2 is the vector (1/2)[v]1.

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An inconsistent system has more than one solution.

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An nxn determinant is defined by determinants of (n-1)x(n-1) submatrices

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An nxn matrix with n linearly independent eigenvectors is invertible.

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An orthogonal matrix is invertible.

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Analogue signals are used in the major control systems for the space shuttle.

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Any linear combination of vectors can always be written in the form A[x] for a suitable matrix A and vector [x].

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Any list of five real numbers is a vector in R5.

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Any solution of (A^t)A[x] = (A^t)[b] is a least-squares solution of A[x] = [b].

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Any system of n linear equations in n variables can be solved by Cramer's rule

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Asking whether the linear system corresponding to an augmented matrix [ [a]1 [a]2 [a]3 [b] ] has a solution amounts to asking whether [b] is in Span {[a]1, [a]2, [a]3}.

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ColA is the set of all solutions of A[x] = [b].

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ColA is the set of all vectors that can be written as A[x] for some [x].

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Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.

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Each eigenvalue of A is also an eigenvalue of A^2.

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Each eigenvector of A is also an eigenvector of A^2.

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Each eigenvector of an invertible matrix A is also an eigenvector of A^-1.

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Each elementary matrix is invertible.

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Eigenvalues must be nonzero scalars.

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Eigenvectors must be nonzero vectors.

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Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

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Every elementary row operation is reversible.

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Every linear transformation is a matrix transformation.

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Every matrix equation A[x] = [b] corresponds to a vector equation with the same solution set.

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Every matrix transformation is a linear transformation.

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Every square matrix is a product of elementary matrices.

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Finding a parametric description of the solution set of a linear system is the same as solving the system.

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For a square matrix A, vectors in Col A are orthogonal to vectors in NulA.

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For an mxn matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A.

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For any scalar c, ||c[v]|| = c||[v]||.

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For each [y] and each subspace W, the vector [y] - the projection of [y] onto W is orthogonal to W.

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If A = QR, where Q has orthonormal columns, then R = (Q^t)A.

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If A = [ A1 A2 ] and B = [ B1 B2 ], with A1 and A2 the same sizes as B1 and B2, respectively, then A+B = [ A1+B1 A2+B2 ].

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If A and B are 2x2 with columns a1, a2, and b1, b2, respectively, then AB = [ a1b1 a2b2 ].

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If A and B are 3x3 and B = [ b1 b2 b3 ], then AB = [ Ab1 + Ab2 + Ab3 ].

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If A and B are invertible nxn matrices, then AB is similar to BA.

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If A and B are mxn, then both A(B^t) and (A^t)B are defined.

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If A and B are nxn and invertible, then (A^-1)(B^-1) is the inverse of AB.

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If A and B are nxn matrices, with detA = 2 and detB = 3, then det(A+B) = 5

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If A and B are nxn, then (A+B)(A-B) = A^2 - B^2.

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If A and B are row equivalent, then their row spaces are the same.

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If A and B are square and invertible, then AB is invertible, and (AB)^-1 = (A^-1)(B^-1).

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If A can be row reduced to the identity matrix, then A must be invertible.

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If A contains a row or column of zeros, then 0 is an eigenvalue of A.

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If A has a QR factorization, say A = QR, then the best way to find the least-squares solution of A[x] = [b] is to compute x-hat = (R^-1)(Q^t)[b].

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If A is a 2x2 matrix with a zero determinant, then one column of A is a multiple of the other

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If A is a 3x2 matrix, then the transformation [x] -> A[x] cannot be one-to-one.

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If A is a 3x2 matrix, then the transformation [x] -> A[x] cannot map R2 onto R3.

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If A is a 3x3 matrix and the equation A[x] = (1 0 0) has a unique solution, then A is invertible.

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If A is a 3x3 matrix with three pivot positions, there exist elementary matrices E1, ... , Ep such that Ep ... E1A = I.

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If A is a 3x3 matrix, then det(5A) = 5detA

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If A is a 3x5 matrix and T is a transformation defined by T([x]) = A[x], then the domain of T is R3.

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If A is an invertible nxn matrix, then the equation A[x] = [b] is consistent for each [b] in Rn.

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If A is an mxn matrix and if the equation A[x] = [b] is inconsistent for some [b] in Rm, then A cannot have a pivot position in every row.

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If A is an mxn matrix whose columns do not span Rm, then the equation A[x] = [b] is inconsistent for some [b] in Rm.

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If A is an nxn matrix, then the equation A[x] = [b] has at least one solution for each [b] in Rn.

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If A is diagonalizable, then the columns of A are linearly independent.

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If A is invertible and 1 is an eigenvalue for A, then 1 is also an eigenvalue for A^-1.

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If A is invertible and if r does not equal zero, then (rA)^-1 = r(A^-1).

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If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In.

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If A is invertible, then the inverse of A^-1 is A itself.

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If A is mxn and the linear transformation [x] -> A[x] is onto, then rankA = m.

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If A is row equivalent to the identity matrix I, then A is diagonalizable.

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If A is similar to a diagonalizable matrix B, then A is also diagonalizable.

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If AB = BA and if A is invertible, then (A^-1)B = B(A^-1).

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If AB = C and C has 2 columns, then A has 2 columns.

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If AB = I, then A is invertible.

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If AC = 0, then either A=0 or C=0.

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If A^3 = 0, then detA = 0

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If A^t is not invertible, then A is not invertible.

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If B = {[b]1, ... , [b]n} and C = {[c]1, ... , [c]n} are bases for a vector space V, then the jth column of the change-of-coordinates matrix P(C<-B) is the coordinate vector [[c]j]b.

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If B is an echelon form of a matrix form of a matrix A, then the pivot columns of B form a basis for ColA.

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If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for RowA.

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If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

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If B is formed by adding to one row of A a linear combination of the other rows, then detB = detA

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If B is obtained from a matrix A by several elementary row operations, then rankB=rankA.

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If B is produced by interchanging two rows of A, then detB = detA

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If B is produced by multiplying row 3 of A by 5, then detB = 5detA

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If B is the standard basis for Rn, then the B-coordinate vector of an [x] in Rn is [x] itself.

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If H = Span { [b]1, ... , [b]p }, then { [b]1, ... , [b]p } is a basis for H.

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If H is a subspace of R3, then there is a 3x3 matrix A such that H = ColA.

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If S is linearly independent, then S is a basis for V.

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If SpanS=V, then some subset of S is a basis for V.

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If T: R2 -> R2 rotates vectors about the origin through an angle, then T is a linear transformation.

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If W = Span {[x]1, [x]2, [x]3} with {[x]1, [x]2, [x]3} linearly independent, and if {[v]1, [v]2, [v]3} is an orthogonal set in W, then {[v]1, [v]2, [v]3} is a basis for W.

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If W is a subspace of Rn and if [v] is in both W and W perp, then [v] must be the zero vector.

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If [b] is in the column space of A, then every solution of A[x] = [b] is a least-squares solution.

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If [u] and [v] are in R2 and det[ [u] [v] ] = 10, then the are of the triangle in the plane with vertices at [0], [u], and [v] is 10

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If [u] is a vector in a vector space V, then (-1)[u] is the same as the negative of [u].

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If [x] and [y] are linearly independent, and if [z] is in Span {[x], [y]}, then {[x], [y], [z]} is linearly dependent.

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If [x] and [y] are linearly independent, and if {[x], [y], [z]} is linearly dependent, then [z] is in Span {[x], [y]}.

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If [x] is in V and if B contains n vectors, then the B-coordinate vector of [x] is in Rn.

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If [x] is not in a subspace W, then [x] - the projection of [x] onto W is not zero.

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If [x] is orthogonal to every vector in a subspace W, then [x] is in W perp.

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If a set S = {[u]j, ... , [u]p} has the property that [u]j*[u]j = 0 whenever i does not equal j, then S is an orthonormal set.

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If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.

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If a set in Rn is linearly dependent, then the set contains more vectors than there are entries in each vector.

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If a set {[v]1, ... , [v]p} spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.

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If an mxn matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of A[x]=[0] is m-k.

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If an nxp matrix U has orthonormal columns, then U(U^t)[x] = [x] for all [x] in Rn.

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If detA is zero, then rows or two columns are the same, or a row or a column is zero

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If dimV = n and S is a linearly independent set in V, then S is a basis for V.

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If dimV = n and if S spans V, then S is a basis for V.

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If dimV = p, then there exists a spanning set of p+1 vectors in V.

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If dimV=p and SpanS=V, then S cannot be linearly dependent.

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If each vector [e]j in the standard basis for Rn is an eigenvector of A, then A is a diagonal matrix.

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If every set of p elements in V fails to span V, then dimV is greater than p.

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If f is a function in the vector space V of all real-valued functions on R and if f(t) = 0 for some t, then f is the zero vector in V.

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If matrices A and B have the same reduced echelon form, then RowA = RowB.

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If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.

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If p is greater than or equal to 2 and dimV = p, then every set of p-1 nonzero vectors is linearly independent.

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If the augmented matrix [ A [b] ] has a pivot position in every row, then the equation A[x] = [b] is inconsistent.

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If the columns of A are linearly dependent, then detA = 0

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If the columns of A are linearly independent, then the columns of A span Rn.

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If the columns of A are linearly independent, then the equation A[x] = [b] has exactly one least-squares solution.

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If the columns of A span Rn, then the columns are linearly independent.

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If the columns of an mxn matrix A are orthonormal, then the linear mapping [x] |-> A[x] preserves length.

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If the columns of an mxn matrix A span Rm, then the equation A[x] = [b] is consistent for each [b] in Rm.

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If the columns of an nxp matrix U are orthonormal, then U(U^t)[y] is the orthogonal projection of [y] onto the column space of U.

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If the distance from [u] to [v] equals the distance from [u] to -[v], then [u] and [v] are orthogonal.

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If the equation A[x] = [0] has a nontrivial solution, then A has fewer than n pivot positions.

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If the equation A[x] = [0] has only the trivial solution, then A is row equivalent to the nxn identity matrix.

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If the equation A[x] = [b] has at least one solution for each [b] in Rn, then the solution is unique for each [b].

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If the equation A[x] = [b] is consistent, then ColA is Rm.

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If the equation A[x] = [b] is inconsistent, then [b] is not in the set spanned by the columns of A.

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If the linear transformation [x] -> A[x] maps Rn into Rn, then A has n pivot positions.

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If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.

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If there exists a linearly dependent set {[v]1, ... , [v]p} in V, then dim V is less than or equal to p.

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If there exists a linearly independent set {[v]1, ... , [v]p} in V, then dimV is greater than or equal to p.

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If there exists a set {[v]1, ... , [v]p} that spans V, then dimV is less than or equal to p.

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If there is a [b] in Rn such that the equation A[x] = [b] is inconsistent, then the transformation [x] -> A[x] is not one-to-one.

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If there is an nxn matrix D such that AD = I, then there is also an nxn matrix C such that CA = I.

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If two row interchanges are made in succession, then the new determinant equals the old determinant

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If two rows of a 3x3 matrix A are the same, then detA = 0

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If vectors [v]1, ... , [v]p span a subspace W and if [x] is orthogonal to each [v]j for j = 1, ... , p, then [x] is in W perp.

F

If x-hat is a least-squares solution of A[x] = [b], then x-hat = [((A^t)A)^-1](A^t([b].

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If {[v]1, ... , [v](p-1)} is linearly independent, then so is {[v]1, ... , [v]p}.

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If {[v]1, ... , [v](p-1)} spans V, then {[v]1, ... , [v]p} spans V.

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If {[v]1, [v]2, [v]3} is an orthogonal basis for W, then multiplying [v]3 by a scalar c gives a new orthogonal basis {[v]1, [v]2, c[v]3}.

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If ||[u]||^2 + ||[v]||^2 = ||[u]+[v]||^2, then [u] and [v] are orthogonal.

T

In a QR factorization, say A = QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A.

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In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.

F

In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.

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In some cases, a plane in R3 can be isomorphic to R2.

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In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.

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In the Orthogonal Decomposition Theorem, each term in formula (2) for y-hat is itself an orthogonal projection of [y] onto a subspace of W.

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Left-multiplying a matrix B by a diagonal matrix A, with nonzero entries on the diagonal, scales the rows of B.

F

Not every linear transformation from Rn to Rm is a matrix transformation.

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Not every linearly independent set in Rn is an orthogonal set.

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Not every orthogonal set in Rn is linearly independent.

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NulA is the kernel of the mapping [x] -> A[x].

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On a computer, row operations can change the apparent rank of a matrix.

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R2 is a subspace of R3.

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R2 is a two-dimensional subspace of R3.

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Reducing a matrix to echelon form is called the forward phase of the row reduction process.

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Row operations on a matrix A can change the linear dependence relations among the rows of A.

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Row operations on a matrix can change the null space.

F

Row operations preserve the linear dependence relations among the rows of A.

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Similar matrices always have exactly the same eigenvalues.

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Similar matrices always have exactly the same eigenvectors.

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The (i,j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its ith row and jth column

T

The Gram-Schmidt process produces from a linearly independent set {[x]1, ... [x]p} an orthogonal set {[v]1, ... , [v]p} with the property that for each k, the vectors [v]1, ... , [v]k span the same subspace as that spanned by [x]1, ... , [x]k.

F

The best approximation to the [y] by elements of a subspace W is given by the vector [y] - the projection of [y] onto W.

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The codomain of the transformation [x] -> A[x] is the set of all linear combinations of the columns of A.

F

The cofactor expansion of detA down a column is the negative of the cofactor expansion along a row

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The column space of A is the range of the mapping [x] -> A[x].

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The column space of an mxn matrix is in Rm.

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The columns of P(C<-B) are linearly independent.

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The columns of a matrix A are linearly independent if the equation A[x] = [0] has the trivial solution.

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The columns of an invertible nxn matrix form a basis for Rn.

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The columns of any 4x5 matrix are linearly dependent.

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The columns of the change-of-coordinates matrix P(C<-B) are B-coordinate vectors of the vectors in C.

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The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the nxn identity matrix.

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The correspondence [[x]] -> [x] is called the coordinate mapping.

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The definition of the matrix-vector product A[x] is a special case of block multiplication.

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The determinant of A is the product of the diagonal entries in A

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The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U

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The determinant of a triangular matrix is the sum of the entries on the main diagonal

F

The equation A[x] = [b] is consistent if the augmented matrix [ A [b] ] has a pivot position in every row.

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The equation A[x] = [b] is homogeneous if the zero vector is a solution.

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The equation A[x] = [b] is referred to as a vector equation.

F

The equation [x] = [p] + t[v] describes a line through [v] parallel to [p].

F

A vector is an arrow in three-dimensional space.

T

AB + AC = A(B+C)

T

The equation [x] = x2[u] + x3[v], with x2 and x3 free (and neither [u] nor [v] a multiple of the other), describes a plane through the origin.

T

The first entry in the product A[x] is a sum of products.

T

The matrices A and A^t have the same eigenvalues, counting multiplicities.

F

The nonpivot columns of a matrix are always linearly dependent.

F

The nonzero rows of a matrix A form a basis for RowA.

F

The normal equations always provide a reliable method for computing least-squares solutions.

F

The rank of a matrix equals the number of nonzero rows.

F

The row reduction algorithm applies only to augmented matrices for a linear system.

T

The row space of A is the same as the column space of A^t.

T

The row space of A^t is the same as the column space of A.

T

The second row of AB is the second row of A multiplied on the right by B.

F

The set Span {[u], [v]} is always visualized as a plane through the origin.

T

The set of all linear combinations of [v]1, ... , [v]p is a vector space.

T

The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.

F

The solution set of A[x] = [b] is obtained by translating the solution set of A[x] = [0].

T

The solution set of the linear system whose augmented matrix is [ [a]1 [a]2 [a]3 [b] ] is the same as the solution set of the equation x1[a]1 + x2[a]2 + x3[a]3 = [b].

F

The standard method for producing a spanning set for NulA, described in Section 4.2, sometimes fails to produce a basis for NulA.

F

The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

F

The sum of two eigenvectors of a matrix A is also an eigenvector of A.

T

The superposition principle is a physical description of a linear transformation.

F

The transpose of a product of matrices equals the product of their transposes in the same order.

T

The transpose of a sum of matrices equals the sum of their transposes.

T

The transpose of an elementary matrix is an elementary matrix.

T

The vector [b] is a linear combination of the columns of a matrix A if and only if the equation A[x] = [b] has at least one solution.

T

The vector [u] results when a vector [u] - [v] is added to the vector [v].

F

The vector spaces P3 and R3 are isomorphic.

F

The weights c1, ... , cp in a linear combination c1[v]1 + ... + cp[v]p cannot all be zero.

T

There exists a 2x2 matrix that has no eigenvectors in R2.

T

A subspace is also a vector space.

F

(AB)C = (AC)B

F

(AB)^t = (A^t)(B^t)

F

A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) [u], [v], and [u] + [v] are in H, and (iii) c is a scalar and c[u] is in H.

T

det((A^t)A) is greater than or equal to 0

F

det(-A) = -detA

F

det(A+B) = detA + detB

F

det(A^t) = (-1)detA

F

det(A^t) = -detA


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