Question

For parts a through f., A denotes an mxn matrix. Determine whether each statement is true or false. Justify each answer. a. A null space is a vector space. Is this statement true or false?
A. True because the null space of an mx n matrix A is a subspace of Rm
B. False, a column space is a vector space, but a null space is not a vector space
C. False, a vector space is a null space, but a null space is not necessarily a vector space
D. True because the null space of an mxn matrix A is a subspace of Rn

Answer 1

Answer:

D. True because the null space of an m x n matrix A is a subspace of $R^n$

Step-by-step explanation:

A null space is also  the vector space.

We know that a null set satisfies the following properties of a vector space.

Now let $x,y \in Null (A), \alpha \in IR$ ,  then

$A[\alpha x+y] = \alpha A(x)+ A(y) = \alpha . 0 + 0 = 0$

Thus, $\alpha x+y \in Null (A)$

Hence, option (d) is true.