4.3

Question 1

T or F: A linearly independent set in a subspace H is a basis for H.

Answer 1

The statement is false because the subspace spanned by the set must also coincide with H.

Question 2

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

Answer 2

The statement is true by the Spanning Set Theorem.

Question 3

T or F: A basis is a linearly independent set that is as large as possible.

Answer 3

The statement is true by the definition of a basis.

Question 4

T or F:The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.

Answer 4

The statement is false because the method always produces an independent set.

Question 5

T or F: If B is an echelon form of a matrix​ A, then the pivot columns of B form a basis for Col A.

Answer 5

The statement is false because the columns of an echelon form B of A are not necessarily in the column space of A.

Question 6

Definition of Basis

Answer 6

Let H be a subpsace of a vector space V. A set of vectors B in V is a basis for H if

i. B is a linearly indepedent set, and
ii. the subspace spanned by B coincides with H; that is,

H = Span B

Question 7

What is Spanning Set Thrm?

Answer 7

Let S = {v_1,….,v_p} be a set in a vector space V, and let H = Span {v_1,….,v_p}.

a. If one of the vectors in S - say, v_k - is a linear combination of the remaining vectors in S, then the set formed from S by removing v_k still spans H.
b. If H doesn’t equal {0}, some subset of S is a basis for H.

Question 8

Thrm: the _______ _______ of a matrix A form a basis for Col A

Answer 8

pivot columns

Question 9

Thrm: If two matrices A & B are ___ __________, then their row spaces are the same. If B is in echelon form, the non zero rows of B form a basis for the row space of A as well as for that of B.

Answer 9

row equivalent