(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let S = [x y z w] [itex]\in[/itex] [tex]R^4[/tex] , 2x-y+2z+w=0and3x-z-w=0

Find a basis for S.

2. Relevant equations

3. The attempt at a solution

I started by putting the system into reduced row form:

[2 -1 2 1]

[3 0 -1 -1]

[2 -1 2 1]

[0 3 -8 -5]

[6 0 -2 -2]

[0 3 -8 -5]

[1 0 -1/3 -1/3]

[0 1 -8/3 -5/3]

Now have:

x - 1/3z - 1/3w = 0

y - 8/3z - 5/3w = 0

Letting z = s, and w = t, we get:

x = 1/3s + 1/3t

y = 8/3s + 5/3t

z = s

w = t

And this gives:

s[1/3 8/3 1 0] and t[1/3 5/3 0 1]

Where the basis vectors are:

[1/3 8/3 1 0] and [1/3 5/3 0 1]

and are linearly independent.

Did I do this correctly? I'm really struggling with these concepts and I feel like I'm missing something. Thanks in advance.

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# LA: Finding a basis

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