1 a Carefully define a subspace S of a vector space V b If A

1. a) Carefully define a subspace S of a vector space V.

b) If A is a mxn matix, then show that N(A) is a vector subspace.

Solution

a) Subspace S of a vector space V - It is defined as say a subset S of a vector space V is a subspace of V if S is a vector space under the inherited addition and scalar multiplication operations of V .

b) Proof - If A is a mxn matix, then show that N(A) is a vector subspace.

first theorem states that if A is a m X n matrix, then S(A) is a subspace of S^m.

Now Null space of A is denoted as N(A). Now we have to prove that N(A) is a vector subspace of Sⁿ.

Firstly notice that if X is in N(A) then AX=0_m. Since A is m X n and AX is m X 1, therefore X must be n X 1.

That is X is in Sⁿ. Hence N(A) is a subspace of Sⁿ.

Now just verify the subspace requirements.