Let A= [1 -2 -3 0
0 1 0 -1
1 -1 1 0]
and K={(x,y,z,w) E R^4: Av^t =0}
Prove that K is a subspace of R^4. Hint: use a matrix equation rather than 3 linear equations.
Any help is greatly appreciated!!
See Linear subspace. $\displaystyle K$ is a subspace because these three conditions hold:
$\displaystyle A0 = 0,$
$\displaystyle Av = 0 \text{ and } Aw = 0 \text{ imply }A(v+w) = Av + Aw = 0,$
$\displaystyle Av = 0 \text{ implies } A(av) = a(Av) = 0 \text{ for scalar } a.$
To show $\displaystyle K$ is a subspace, you must show the three conditions listed in the linear subspace link hold.
The first condition, for example, is $\displaystyle 0 \in K.$ To show this holds, you note as I did that $\displaystyle A0 = 0$ so the vector $\displaystyle 0$ satisfies the equation defining $\displaystyle K$, which is $\displaystyle Av = 0$. Notice the hint says use a matrix equation.
The other two conditions are done similarly.