LINEAR ALGEBRA EXAM 2 CONCEPT PROBLEMS QUESTIONS WITH CORRECT ANSWERS

LINEAR ALGEBRA EXAM 2 CONCEPT
PROBLEMS QUESTIONS WITH
CORRECT ANSWERS
if A is an nxn matrix and the columns of A span Rn then Ax=0 has only the trivial
solution - Answer-true

if A is a 6x7 matric and the null space of A has dimension 4, then the column space of A
is a 2-plane - Answer-false

if A is an mxn matrix and m>n then the linear transformation T(x)=Ax cannot be one-to-
one - Answer-false

if A is an nxn matrix and its rows are linearly independent the Ax=b has a unique
solution for every b in Rn - Answer-true

the solution set of a consistent matrix equation Ax=b is a subspace - Answer-false

there exists a 3x5 matrix with rank 4 - Answer-false

if A is a 2x5 matrix then NulA is a subspace of R2 - Answer-false

if A has more columns than rows then T is not onto - Answer-false

if A is an nxn matrix and Ax=0 has only the trivial solution then the equation Ax=b is
consistent for every b in Rn - Answer-true

if a matrix A has more columns than rows then the linear transformation T given by
T(x)=Ax is not one-to-one - Answer-true

if A and B are 3x3 matrices and the columns of B are linearly dependent then the
columns of AB are linearly dependent - Answer-true

if a set S of vectors contains fewer vectors than there are entries in the vectors then the
set is linearly independent - Answer-false

if a matrix A has more rows then columns then the linear transformation T given by
T(x)=Ax is not onto - Answer-true

suppose v1, v2, v3 are vectors in R4. if {v1,v2} is linearly independent and v3 is not in
span{v1,v2} then {v1,v2,v3} must be linearly independent - Answer-true

, suppose that V is a 2D subspace of R3 and that (1,3,-1) and (0,1,2) are in V. then {(1,3,-
1),(0,1,2)} is a basis for V - Answer-true

if A is a 3x3 matrix then ColA must contain the vector (0,0,0) - Answer-true

suppose T: Rn to Rm is a linear transformation with standard matrix A. if T is not one-to-
one then Ax=0 must have infinitely many solutions - Answer-true

let A be a 4x6 matric and let T be the matrix transformation T(x)=Ax. which of the
following are possible?
a) NulA is a line through the origin
b) for every b in R4 the equation Ax=b is consistent
c) dim(ColA)=6
d) for some b in R4 the equation T(x)=b has a unique solution
e) for every b in R4 the equation T(x)=b has at most one solution - Answer-b

if {v1,v2,v3,v4} is a basis for a subspace V of Rn the {v1,v2,v3} is a linearly independent
set - Answer-true

if A is a 9x4 matrix with a pivot in each column then NulA={0} - Answer-true

if A is a matrix with more rows than columns then the transformation T(x)=Ax is not one-
to-one - Answer-false

a translate of a span is a subspace - Answer-false

there exists a 4x7 matrix A such that nullity A=5 - Answer-true

if {v1,v2,...,vn} is a basis for R4 then n=4 - Answer-true

which of the following are onto transformations?
a) T: R3 to R3 reflect over xy-plane
b) T: R3 to R3 project onto xy-plane
c) T: R3 to R2 project onto xy-plane forget z coordinate
d) T: R2 to R2 scale the x-direction by 2 - Answer-a, c, d

let A be a square matrix and let T(x)=Ax, which guarantee T is onto?
a) T is one-to-one
b) Ax=0 is consistent
c) ColA=Rn
d) there is a transformation U such that T(U(x))=x for all x - Answer-a, c, d

if A is a 5x3 matrix and B is a 4x5 matrix then the transformation T(x)=BAx has domain
R3 and codomain R4 - Answer-true

there is a 4x7 matrix A that satisfies dim(NulA)=1 - Answer-false