Are null space vectors unique?

Question

If I have a matrix $A$, where $$A = \pmatrix{2 & 2 \\ 5 & 5}$$

Would the null space vector be $\langle-1,1\rangle$ or $\langle 1,-1\rangle $ or are they both valid null space vectors?

Answer 1

A vector $v$ is in the null space of a matrix $A$ if $Av = 0$. So if $v$ is a non-zero vector in the null space of $A$, then $$ A(\lambda v) = \lambda (Av) = \lambda \cdot 0 = 0 $$ and so any scalar multiple $\lambda v$ of $v$ is also in the null space. In particular, in your case $v = (1, -1)$ and $-v = (-1, 1)$ are both in the null space.

Answer 2

$\langle 1,-1 \rangle$ is the set of elements of the form $(u,-u)$, for $u \in \mathbb{C}$.

$\langle -1,1 \rangle$ is the set of elements of the form $(-u,u)$, for $u \in \mathbb{C}$.

You see that the two sets are identical.

Answer 3

They would both be the null space because the null space is a subspace. Subspaces are closed under scalar multiplication. Your two answers are just scalar multiples of each other

$\Bigg \langle 1,-1 \Bigg \rangle$ multipled by the scalar -1 is $\Bigg \langle -1,1 \Bigg \rangle$