Linear Algebra
A consistent system of linear equations has exactly one solution.
False
A subset H of a vector space V is a subspace of V if the zero vector is in H
False
When A and B are n × n, invertible matrices, then AB is invertible and (AB)^−1 = (A^−1)(B^−1)
False
When A has a diagonalization P DP^−1 this is a Singular Value Decomposition of A
False
When A is an n×n matrix and the equation Ax = b is consistent, then the solution is unique
False
When u, v are nonzero vectors, then Span{u, v} contains only the line through u and the origin, and the line through v and the origin
False
If A is an n × n matrix, when does the equation Ax = b have at least one solution for each b in Rn
Sometimes
If T : R2 → R2 rotates vectors counterclockwise about the origin through an angle θ, then T is a linear transformation
True
NOT CLOSED UNDER VECTOR ADDITION
True
If u, v, and w are nonzero vectors in R2 and u is not a multiple of v, is w a linear combination of u and v?
Always
A product of invertible n × n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
False
A set B = {v1, v2, . . . , vp} of vectors in Rn is always linearly independent when p < n.
False
A symmetric n × n A matrix always has n distinct real eigenvalues
False
Every matrix is row equivalent to a unique matrix in echelon form.
False
Every orthogonal matrix is orthogonally diagonalizable
False
For n × n matrix A is said to be diagonalizable when A = P DP^−1 for some matrix D and invertible matrix P
False
Four linearly independent vectors in R5 span R5
False
If A and B are 3 × 3 matrices and B = [b1 b2 b3] , then the product AB is given by AB = [Ab1 + Ab2 + Ab3] .
False
If A is a 2×2 symmetric matrix, then the set of x such that x T Ax = c, for some constant c, corresponds to either a circle, an ellipse, or a hyperbola
False
If A is an m × n matrix and the equation Ax = b is consistent for some b in Rm, then the columns of A span Rm
False
If A is an m × n matrix, then the range of the transformation TA : x → Ax is Rm
False
If A is an n × n matrix and Ax = λx for some scalar λ, then x is an eigenvector of A
False
If A is symmetric, then the change of variable x = P y transforms Q(x) = x^T Ax into a quadratic form with no cross-product term for any orthogonal matrix P
False
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A
False
If L is a line through the origin and if projLy is the orthogonal projection of y onto L, then kprojLyk is the distance from y to L
False
If a set S = {u1, ..., up} has the property that ui · uj = 0 whenever i /= j, then S is an orthonormal set
False
If a system of linear equations has no free variables, then it has a unique solution.
False
If an n × n matrix A is diagonalizable, then A has n distinct eigenvalues
False
If e1, e2, and e3 are the standard basis vectors for R3 then B = {e1, e2, e1 − e2, e3} is a basis for R3
False
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
False
If the equation A x = b is consistent, then b is in the set spanned by the rows of A
False
If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.
False
If v1 and v2 are linearly independent eigenvectors of an n × n matrix A, then they correspond to distinct eigenvalues of A
False
If {v1, v2, v3} is an orthogonal basis for W, then multiplying v3 by a scalar c gives a new orthogonal basis {v1, v2, c v3} for W
False
Let V be a vector space. If dim V = n and if S spans V , then S is a basis for V
False
The best approximation to y by elements of a subspace W is given by the vector y − projW y
False
The columns of a matrix A are linearly independent when the equation Ax = 0 has the solution x = 0.
False
The composition of two linear transformations need not be a linear transformation.
False
The determinant of a triangular matrix is always the sum of the entries on the main diagonal
False
The dimension of Nul A is the number of variables in the equation Ax = 0.
False
The dimension of the vector space P4 of all polynomials of degree at most four is 4
False
The eigenvalues of an n × n matrix A are the entries on the main diagonal of A
False
The equality (ABC)^T = C^T A^T B^T holds for all n × n matrices A, B, and C.
False
The homogeneous equation Ax = 0 has the trivial solution x = 0 if and only if the equation has at least one free variable.
False
The orthogonal projection, projW y, of y onto a subspace W depends on the orthogonal basis for W used to compute it
False
The pivot columns of rref(A) form a basis for Col A.
False
The set H of all polynomials p(x) = a + x^4 a in R , is a subspace of the vector space P6 of all polynomials of degree at most 6
False
The singular values of a matrix A are all positive
False
The subset V = a b : ab ≥ 0 of R2 is closed under vector addition.
False
The sum of the vector u − v and the vector v is the vector v.
False
Three vectors in R5 always span R5
False
When u = [−2 5] v = [−5 2] , then the vectors in Span{u, v} lie on a line through the origin.
False
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix
True
A basis {v1, v2, . . . , vp} for a vector space V is a set such that Span{v1, v2, . . . , vp} = V for which p is as small as possible
True
A linear transformation T : Rn → Rm is completely determined by its effect on the columns e1, e2, . . . , en of the n × n identity matrix In.
True
A linear transformation is a function T : Rn → Rm such that T(c1a + c2b) = c1T(a) + c2T(b) for all vectors a, b in Rn and all scalars c1, c2.
True
A quadratic form can always be written as Q(x) = x^T Ax with A a symmetric matrix
True
A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix
True
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) the sum u + v is in H for all u, v in H, iii) the scalar multiple cu is in H for all scalars c and u in H
True
A subspace H of a vector space V is a vector space by itself
True
A transformation T : Rn → Rm is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all vectors v1, v2 in Rn and all scalars c1, c2.
True
A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution.
True
An example of a linear combination of vectors v1 and v2 is the vector −2v1.
True
An n × n matrix A is not invertible if 0 is an eigenvalue of A
True
An n × n matrix that is orthogonally diagonalizable must be symmetric
True
Any list of five real numbers is a vector in R5
True
Every linear transformation T : Rn → Rm is a matrix transformation.
True
Every n × n matrix A having linearly independent eigenvectors v1, v2, . . . , vn can be diagonalized
True
Every symmetric matrix is orthogonally diagonalizable
True
For each fixed 3 × 2 matrix B, the corresponding set H of all 2 × 4 matrices A such that BA = 0 is a subspace of R 2×4
True
For each y in Rn and each subspace W of Rn the vector y − projW y is in W⊥
True
For every matrix equation Ax = b there corresponds a vector equation having the same solution set.
True
Four linearly independent vectors in R4 span R4
True
If A can be row reduced to the identity matrix, then A must be invertible.
True
If A is a 7 × 7 matrix having eigenvalues λ1, λ2, and λ3 such that (i) the eigenspace corresponding to λ1 is two-dimensional, (ii) the eigenspace corresponding to λ2 is three-dimensional, then A need not be diagonalizable.
True
If A is an invertible n × n matrix, then the equation Ax = b is consistent for every b in Rn
True
If A is an m × n matrix, then A^TA is orthogonally diagonalizable.
True
If A is an m × n matrix, then the range of the transformation T : Rn → Rm, TA : x → A x , is the set of all linear combinations of the columns of A
True
If A is an n × n invertible matrix with singular value σ, then A −1 has singular value σ^−1
True
If A is an n × n matrix and its columns are linearly independent, then the columns of A span Rn
True
If A is an n × n matrix, then (A^2)^T = (A^T)^2
True
If A is invertible, then the inverse of A^−1 is A itself.
True
If H is a p-dimensional subspace of Rn then a linearly independent set of p vectors in H is a basis for H.
True
If W is a subspace of Rn and y is a vector in Rn such that y = z1 + z2 with z1 in W and z2 in W⊥, then z1 is the orthogonal projection, projW y, of y onto W
True
If a set of p vectors spans a p-dimensional subspace H of Rn then these vectors form a basis for H
True
If the columns of an n × n matrix A are linearly dependent, then detA = 0.
True
If the columns of an n×n matrix A span Rn then the columns are linearly independent
True
If u and v are linearly independent and w is in Span{u, v}, then the set {u, v, w} is linearly dependent.
True
If u1, u2, u3 and u4 are vectors in R7 and u2 = 0, then the set S = {u1, u2, u3, u4} is linearly dependent.
True
Not every orthogonal set in Rn is linearly independent
True
The Algebraic Multiplicity of an eigenvalue λ of a square matrix A is the multiplicity of λ as a root of the characteristic equation of A
True
The determinant of an n × n matrix A can be defined recursively in terms of the determinants of (n − 1) × (n − 1) submatrices of A
True
The equation Ax = b is homogeneous if the zero vector is a solution.
True
The expression |x|^2 is a quadratic form
True
The four points (0, 0, 0), (2, −3, −12), (−1, −2, −1), (−3, −1, 7), in 3-space are co-planar, i.e., lie on a plane.
True
The only three-dimensional subspace of R3 is R3 itself
True
The product of the m × n matrix A = [a1 a2 . . . an] and the vector x = [ x1 x2 xn ] in Rn is the vector x1a1 + x2a2 + . . . + xnan in Rm
True
There are 2 × 2 matrices A, B, and C for which AB = AC and A =/ 0, but B /= C
True
When A is symmetric, then an orthogonal diagonalization of A need not be a Singular Value Decomposition of A
True
When u, v are vectors in Rn such that dist(u, v) = dist(u, −v), then u, v are orthogonal
True
is a vector space under the usual addition and scalar multiplication of vectors in R3
True
