I-am-learning-linear-algebra-and-have-a-qustion-in-regards-to-nullspace-So-Ax-0-If-A-1-1-0-0-this-is-simply-solved-for-x-x-1-x-2-by-1-1-0-0-0-0-x-1-1-c-c-

Question Number 124170 by Pengu last updated on 01/Dec/20

I am learning linear algebra and have a

qustion in regards to nullspace .

So, \boldsymbol A x = 0 .

If \boldsymbol A = [\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}], this is simply solved for

\boldsymbol x = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]

by :

[\begin{matrix} 1 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] \to \boldsymbol x = [\begin{matrix} - 1 \\ 1 \end{matrix}] c, \forall c \in R

Since this is for all \boldsymbol x \in R^{2}, the null space

falls on a line (in this case) .

My question is, in higher dimensions,

can the null space be any dimension up

to the dimension n ? So, in R^{4}, can the

null space be a point, line, plane, or volume ?

Furthermore, can you have m u l t i p l e variations

of planes, points, etc . that make up the null space ?

Thanks!