Linear Algebra Exam 2 Concept Problems
if A is a 2x5 matrix then NulA is a subspace of R2
false
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
false
if A is a matrix with more rows than columns then the transformation T(x)=Ax is not one-to-one
false
if A is an mxn matrix and m>n then the linear transformation T(x)=Ax cannot be one-to-one
false
if T: Rn to Rm is a linear transformation and n<m then the equation T(x)=0 must have infinitely many solutions
false
if a matrix A has more rows than columns then the matrix transformation T(x)=Ax is not one-to-one
false
if a set S of vectors contains fewer vectors than there are entries in the vectors then the set is linearly independent
false
the solution set of a consistent matrix equation Ax=b is a subspace
false
let A be an mxn matrix and let T(v)=Av be the associated linear transformation suppose T is not one-to-one, must it be true that Ax=0 has infinitely many solutions?
yes
let V be the set of solutions to x+y+z=0 in R3 is V subspace of R3?
yes
say that V is a plane in R3, that v and w are two vectors in V, and that neither v or w is a multiple of the other must it be true that {u,v} is a basis for V?
yes
suppose that A is a 2x3 matrix and the column space for A is a plane is it possible for Ax=b to have infinitely many solutions?
yes
suppose that {u,v} is a basis for a subspace V of R3 must it be true that {u+v,v} is a basis for V?
yes
suppose we have a set of 100 vectors in R99 must it be true that the set is linearly dependent?
yes
a translate of a span is a subspace
false
if A has more columns than rows then T is not onto
false
suppose A is an mxn matrix and m<n then the matrix transformation T(x)=Ax is not onto
false
suppose v1, v2, v3, v4 are vectors in R5 so that span{v1,v2} has dimension 2 and span{v3,v4} has dimension 2. then span{v1,v2,v3,v4} has dimension 4
false
there exists a 3x5 matrix with rank 4
false
there is a 4x6 matrix whose rank is 5 and whose nullity is 1
false
there is a 4x7 matrix A that satisfies dim(NulA)=1
false
T: R to R given by T(x)=x+1, is T a linear transformation?
no
is it possible for a 3x5 matrix to have the dimensions of its column space and null space be equal?
no
if A and B are 3x3 matrices and the columns of B are linearly dependent then the columns of AB are linearly dependent
true
if A is a 3x3 matrix and Ax=(1,0,1) has exactly one solution then every vector in R3 is in the span of the columns of A
true
if A is a 3x3 matrix then ColA must contain the vector (0,0,0)
true
if A is a 4x5 matrix and the solution set to Ax=0 is a line then the matrix transformation T(x)=Ax is onto
true
if A is a 5x3 matrix and B is a 4x5 matrix then the transformation T(x)=BAx has domain R3 and codomain R4
true
if A is a 9x4 matrix with a pivot in each column then NulA={0}
true
if A is a kx5 matrix and the columns of A form a basis for Rk then k=5
true
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
true
if A is an nxn matrix and its rows are linearly independent the Ax=b has a unique solution for every b in Rn
true
if A is an nxn matrix and the columns of A span Rn then Ax=0 has only the trivial solution
true
if T: Rn to Rm is a one-to-one linear transformation and m does not equal n the T must not be onto
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
true
if a matrix A has more rows then columns then the linear transformation T given by T(x)=Ax is not onto
true
if the column vectors of a 3x3 matrix A span R3 then A has 3 pivots
true
if u,v and w form a basis of subspace W then u+v, v and w also form a basis for W
true
if {v1,v2,...,vn} is a basis for R4 then n=4
true
if {v1,v2,v3,v4} is a basis for a subspace V of Rn the {v1,v2,v3} is a linearly independent set
true
if {v1,v2,v3,v4} is a linearly independent set of vectors in R4 then {v1,v2,v3,v4} must be a basis for R4
true
suppose A is a 3x3 matrix and the vector (1,0,0) is not in ColA then the transformation T(x)=Ax cannot be one-to-one
true
suppose A is a 4x5 matrix and the column span of A has dimension 2 then the set of solutions to Ax=0 is a 3D subspace of R5
true
suppose A is a matrix with more columns than rows, then the matrix transformation T(x)=Ax cannot be one-to-one
true
suppose A is an nxn matrix and Ax=0 has only the trivial solution, then each b in Rn can be written as a linear combination of the columns of A
true
suppose A is an nxn matrix and the matrix transformation T given by T(x)=Ax is onto, then T must also be one-to-one
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
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
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
true
there are linear transformations T: R4 to R3 and U: R3 to R4 so that T(U) is onto
true
there exists a 4x7 matrix A such that nullity A=5
true
there is a 5x4 matrix whose rank is 2 and whose nullity is 2
true
suppose A is a 7x4 matrix such that the associated linear transformation T: R4 to R7 is one-to-one what is rank(A)?
4
consider the plane z=1 in R3, which properties of a subspace are failed by V? a) zero vector: the zero vector is in V b) closure under addition: if u and v in V the u+v is in V c) closure under scalar multiplication: if u is in V and c is a scalar the cu is in V d) none of the above, V is a subspace
a, b, c
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
a, c, d
suppose A is a 11x3 matrix, B is a 3x4 matrix, and C is a 4x11 matrix which of the following matrix multiplications is allowed? a) AB b) AC c) BC d) CA
a, c, d
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
a, c, d
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
b
suppose that A is a 2x3 matrix and that the linear transformation T(v)=Av is onto, describe the solutions of Ax=0 a) a line in R2 b) R2 c) a line in R3 d) a plane in R3 e) none of the above
c
suppose that T: Rn to Rm is a linear transformation with standard matrix A, which of the following conditions guarantee that T is one-to-one? a) for each x in Rn there is a unique y in Rm so that T(x)=y b) for each y in Rm the matrix Ax=y is consistent c) the columns fo A are linearly independent
c
