Linear Algebra True/False

1.3: 23 a) Another notation for the 'vertical' vector (-4,3) is 'horizontal' (-4,3) b) The points in the plane corresponding to (-2,5) and (-5,2) lie on a line through the origin c) An example of a linear combination of vectors v1 and v2 is the vector 1/2(v1) d) The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + a3x3 = b e) The set Span {u, v} is always visualized as a plane through the origin

a) False b) False c) True d) True e) False

1.2: 21 a) In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations b) The row reduction algorithm applies only to augmented matrices for a linear system c) A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix d) Finding a parametric description of the solution set of a linear system is the same as SOLVING the system e) If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent

a) False b) False c) True d) True e) False

2.8: 22 a) A subset of H of R^n is a subspace if the zero vector is in H b) If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A c) Given vectors v2,...,vp in R^n, the set of all linear combinations of these vectors is a subspace of R^n d) Let H be a subsapce of R^n, If x is in H, and y is in R^n, then x + y is in H e) The column space of a matrix A is the set of solutions of Ax = b

a) False b) False c) True d) False e) False

1.7: 21 a) The columns of a matrix A are linearly independent if ht equation Ax = 0 has the trivial solution b) If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S c) The columns of any 4x5 matrix are linearly dependent d) If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span{x,y}

a) False b) False c) True d) True

2.1: 15 a) If A and B are 2x2 matrices with columns a1, a2, and b1, b2, respectively, then AB = [a1b1 a2b2] b) Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A c) AB + AC = A(B + C) d) A^T + B^T = (A+B)^T e) The transpose of a product of matrices equals the product of their transposes in the same order

a) False b) False c) True d) True e) False

1.5: 24 a) A homogeneous system of equations can be inconsistent b) If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero c) The effect of adding p to a vector is to move the vector in a direction parallel to p d) The equation Ax = b is homogeneous if the zero vector is a solution e) If Ax = b is consistent, then the solution set of Ax = b is obtained by translating the solution set of Ax = 0

a) False b) False c) True d) True e) True

1.1: 24 a) Two matrices are row equivalent if they have the same number of rows b) Elementary row operations on an augmented matrix never change the solution set of the associated linear system c) Two equivalent linear systems can have different solution sets d) A consistent system of linear equations has one or more solutions

a) False b) True c) False d) True

1.4: 23 a) The equation Ax = b is referred to as a VECTOR EQUATION b) 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 c) The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row d) The first entry in the product Ax is a sum of products e) If the columns of an mxn matrix A span R^m, then the equation Ax = b is consistent for each b in R^m f) If A is an mxn matrix and if the equation Ax = b is inconsistent for some b in R^m, then A cannot have a pivot position in every row

a) False b) True c) False d) True e) True f) True

2.2: 10 a) If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In b) If A is invertible, then the inverse of A^-1 is A itself c) A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order d) If A is an nxn matrix and Ax = ej is consistent for every j in {1,2,...,n}, then A is invertible. Note: e1,...en represent the columns of the identity matrix e) If A can be row reduced to the identity matrix, then A must be invertible

a) False b) True c) False d) True e) True

1.9: 24 a) If A is a 4x3 matrix, then the transformation x |--> maps R^3 onto R^4 b) Every linear transformation from R^n to R^m is a matrix transformation c) The columns of the standard matrix for a linear transformation from R^n to R^m are the images of the columns the nxn identity matrix under T d) A mapping T: R^n --> R^m is one-to-one if each vector in R^n maps onto a unique vector in R^m e) The standard matrix of a horizontal shear transformation from R^2 to R^2 has the form matrix(a,0,0,d), where a and d are +- 1

a) False b) True c) True d) False e) True

1.5: 23 a) A homogeneous equation is always consistent b) The equation Ax = 0 gives an explicit description of its solution set c) The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable d) The equation x = p + tv describes a line through v parallel to p e) The solution set of Ax = b is the set of all vectors of the form w = p + vh, where vh is any solution of the equation Ax = 0

a) True b) False c) False d) False e) False

1.8: 21 a) A linear transformation is a special type of function b) If A is a 3x5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R^3 c) If A is an mxn matrix, then the range of transformation x |--> Ax is R^m d) Every linear transformation is a matrix transformation e) A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2

a) True b) False c) False d) False e) True

1.2: 22 a) The reduced echelon form of a matrix is unique b) If every column of an augmented matrix contains a pivot, then the corresponding system is consistent c) The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process d) a general solution of a system is an explicit description of all solutions of the system e) Whenever a system has free variables, the solution set contains many solutions

a) True b) False c) False d) True e) False

1.3: 24 a) When u and v are nonzero vectors, Span {u, v} contains only the line through u and the origin, and the line through v and the origin b) Any list of five real numbers is a vector in R^5 c) Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span {a1, a2, a3} d) The vector v results when a vector u-v is added to the vector v e) The weights c1,...,cp in a linear combination c1v1 + ... + cpvp cannot all be zero

a) True b) True c) True d) False e) False

1.4: 24 a) Every matrix equation Ax = b corresponds to a vector equation with the same solution set b) If the equation Ax = b is consistent, then b is in the set spanned by the columns of A c) Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x d) If the coefficient matrix A has a pivot position in every row, then the equation Ax = b is inconsistent e) The solution set of a linear system wwhose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax = b, if A = [a1 a2 a3] f) If A is an mxn matrix whose columns do not span R^m, then the equation Ax = b is consistent for every b in R^m

a) True b) True c) True d) False e) True f) False

1.9: 23 a) A linear transformation T: R^n --> R^m is completely determined by its effect on the columns of the nxn identity matrix b) If T: R^2 --> R^2 rotates vectors about the origin through an angle theta, then T is a linear transformation c) When two linear transformations are performed one after another, the combined effect may not always be a linear transformation d) A mapping T: R^n --> R^m is onto R^m if every vector x in R^n maps onto some vector in R^m e) If A is a 3x2 matrix, then the transformation x |--> Ax cannot be one-to-one

a) True b) True c) False d) False e) False

1.1: 23 a) Every elementary row operation is reversible b) A 5x6 matrix has six rows c) 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. d) two fundamental questions about a linear system involve existence and uniqueness

a) True b) False c) False d) TRUE

2.2: 9 a) In order for a matrix B to be the inverse of A, the equations AB = I and BA = I must both be true b) If A and B are nxn and invertible, then A^-1B^-1 is the inverse of AB c) If A = [a b c d] and ab - cd =/ 0, then A is invertible d) If A is an invertible nxn matrix, then the equation Ax = b is consistent for each b in R^n e) Each elementary matrix is invertible

a) True b) False c) False d) True e) True

2.3: 12 a) If there is an nxn matrix D such that AD = I, then DA = I b) If the linear transformation x |--> Ax maps R^n into R^n, then the row reduced echelon form of A is I c) If the columns of A are linearly independent, then the columns of A span R^n d) If the equation Ax = b has at least one solution for each b in R^n, then the transformation x |--> Ax is not one-to-one e) If there is a b in R^n such that the equation Ax = b is consistent, then the solution is unique

a) True b) False c) Tru d) False e) False

2.1: 16 a) The first row of AB is the first row of A multiplied on the right by B b) If A and B are 3x3 matrices and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3] c) If A is an nxn matrix, then (A^2)^T = (A^t)^2 d) (ABC)^t = C^tA^TB^T e) The transpose of a sum of matrices equals the sum of their transposes

a) True b) False c) True d) False e) True

2.9: 17 a) If Q = {v1,...,vp} is a basis for a subspace H and if x = c1v1 + ... + cpvp, then c1,...,cp are the coordinates of x relative to the basis Q b) Each line in R^n is a one-dimensional subspace of R^n c) The dimension of Col A is the number of pivot columns in A d) The dimensions of Col A and Nul A add up to the number of columns in A e) If a set of p vectors spans a p-dimensional subspace H of R^n, then these vectors form a basis for H

a) True b) False c) True d) True e) True

1.7: 22 a) If u and v are linearly independent, and if w is in Span{u,v}, then {u,v,w} is linearly dependent b) If three vectors in R^3 lie in the same plane in R^3, then they are linearly dependent c) If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent d) If a set in R^n is linearly dependent, then the set contains more than n vectors

a) True b) True c) False d) False

1.8: 22 a) The range of the transformation x |--> Ax is the set of all linear combinations of the columns of A b) Every matrix transformation is a linear transformation c) If T: R^n --> R^m is a linear transformation and if c is in R^m, then a uniqueness question if "Is c in the range of T?" d) A linear transformation preserves the operations of vector addition and scalar multiplication e) A linear transformation T: R^n --> R^m always maps the origin R^n to the origin of R^m

a) True b) True c) False d) True e) True

2.8: 21 a) A subspace of R^n is any set H such that (i) the zero vector is in H, (ii)u, v, and u+v are in H, and (iii) c is a scalar and cu is in H b) If v1,...,vp are in R^n, then Span{v1...v2} is the same as the column space of the matrix [v1...vp] c) The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R^m d) The columns of an invertible nxn matrix form a basis for R^n e) Row operations do not affect linear dependence relations among the columns of a matrix

a) True b) True c) False d) True e) True

2.9: 18 a) If Q is a basis for subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in Q b) The dimension of Nul A is the number of variables in the equation Ax = 0 c) The dimension of the column space of A is rank A d) If Q = {v1,...vp} is a basis for subspace H of R^n, then the correspondence x |--> [x]_Q makes H look and act the same as R^p e) If H is a p-dimensional subspace of R^n, then a linearly independent set of p vectors in H is a basis for H

a) True b) True c) False d) True e) True

2.3: 11 a) If the equatoin Ax = 0 has only the trivial solution, then A is row equivalent to the nxn identity matrix b) If the columns of A span R^n, then the columns are linearly independent c) If A is an nxn matrix, then the equation Ax = b has at least one solution for each b in R^n d) If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions e) If A^T is not invertible, then A is not invertible

a) True b) True c) False d) True e) True