Can you find the null space of this matrix please?

Question

Hey guys,

I am supposed to find a matrix A whose null space is exactly the subspace of R^4 spanned by the following vectors:

Vectors here: http://www.so-easy.info/nullspace_q.gif

I'm having a hard time figuring this out. If you can do it, can you please explain how you did it?

Thank you for your help!

Thank you Pascal! You're the best!

Answer 1

Okay, the first thing we note is that every single one of those vectors is a scalar multiple of (4, 3, 2, 3), so their span is just span {(4, 3, 2, 3)}. So we need a matrix that sends (4, 3, 2, 3) to zero and every vector that is linearly independent from it to a nonzero vector. We note that if v₁, v₂, v₃, and v₄ are the column vectors of A, then A*(4, 3, 2, 3) = 4v₁ + 3v₂ + 2v₃ + 3v₄. To ensure this sum is zero, we may choose the first three vectors arbitrarily and then set v₄ = -1/3 (4v₁ + 3v₂ + 2v₃). Then the kernel of this matrix will be at least span {(4, 3, 2, 3)}. If additionally v₁, v₂, and v₃ form a linearly independent set, then the rank of A will be 3, so the kernel will be at most {(4, 3, 2, 3)}, thus giving us the desired matrix. So choose v₁=e₁, v₂=e₂, and v₃=e₃. Then A becomes:

[1, 0, 0, -4/3]
[0, 1, 0, -1]
[0, 0, 1, -2/3]
[0, 0, 0, 0]

It is easily verified that this matrix works, so we are done.

Answer 2

notice that all the vectors are multiples of the first vector
So the subscpace of R^4 spanned by those vectors is in fact the subspace spanned only by the first vector