1. ## Subspace

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!!

2. Originally Posted by Jacko
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:

$\displaystyle A0 = 0,$

$\displaystyle Av = 0 \text{ and } Aw = 0 \text{ imply }A(v+w) = Av + Aw = 0,$

$\displaystyle Av = 0 \text{ implies } A(av) = a(Av) = 0 \text{ for scalar } a.$

3. Yeah we've been told those conditions in lectures but only been shown basic examples.
Anyone know how to do this question?

4. Originally Posted by Jacko
Yeah we've been told those conditions in lectures but only been shown basic examples.
Anyone know how to do this question?

Don't be scared of it, just show the conditions JakeD has mentioned.

5. Originally Posted by Jacko
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!!
Originally Posted by JakeD
See Linear subspace. $\displaystyle K$ is a subspace because these three conditions hold:

$\displaystyle A0 = 0,$

$\displaystyle Av = 0 \text{ and } Aw = 0 \text{ imply }A(v+w) = Av + Aw = 0,$

$\displaystyle Av = 0 \text{ implies } A(av) = a(Av) = 0 \text{ for scalar } a.$
Originally Posted by Jacko
Yeah we've been told those conditions in lectures but only been shown basic examples.
Anyone know how to do this question?
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.