# Finding a basis of a subspace

• PenTrik

## Homework Statement

Find a basis for each of these subspaces of R4

All vectors that are perpendicular to (1,1,0,0) and (1,0,1,1)

2. The attempt at a solution
I'm not sure how to approach this question. The only thing I can think of is that a vector that would be perpendicular to both would be where the dot product would equal zero aye?

So then that would give me

1x1 + 1x2 = 1x1 + 1x3 + 1x4 = 0.
So in which case, I'd do RREF
[[1,1,0,0]
[1,0,1,1]]

to

[[1,0,1,1]
[0,1,-1,-1]]

I get stuck here because I'm not sure how to solve for the basis at this point.
I'm also unsure if what I did for rref was correct because vectors are usually denoted by column spaces, rather than what row spaces such as what I have done.

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That's the right start. But now you have to solve the linear equations. You've got 2 equations in 4 unknowns. You should be able to express the vector (a,b,c,d) in terms of two parameters by eliminating two unknowns. Can you do that?

I'm not sure if this is what I'm supposed to do but

Given that I have the RREF form, that gives me
x1 + x3 + x4 = 0
x2 - x3 - x4 = 0

Given that x1 and x2 are pivots, this gives me
x1 = -x3 - x4
x2 = x3 + x4

So, if I substitute x3 and x4 with a and b, this makes my basis the span of <-1,1,1,0> and <-1,1,0,1> ?

Well, both of those vectors are perpendicular to the given vectors. And they are linearly independent, so sure, they are a basis. The subspace they span is the perpendicular subspace, the span itself isn't a 'basis'. Those two vectors are a basis.

Ah, your help was much appreciated.