Find a linearly independent set of vectors that spans the same subspace of R4 as that spanned by the vectors

Question

[1 -2 2 -3], [0 3 -1 3], [-2 -1 -6 3], [1 -4 -1 -3]

Answer 1

Let v₁ = <1, -2, 2, -3>, v₂ = <0, 3, -1, 3>, etc

We must find constants c₁, c₂, c₃, c₄ so that c₁v₁ + ... + c₄v₄ = <0, 0, 0, 0>

This amounts to putting the 4 x 4 matrix with first column v₁, second column v₂, etc into row reduced echelon form.

Doing this, we get c₁ = -3c₄, c₂ = -c₄, and c₃ = -c₄. So, letting c₄ = 1 for example, we get

<1, -4, -1, -3> = 3<1, -2, 2, -3> + <0, 3, -1, 3> + <-2, -1, -6, 3>

In other words, v₄ is a linear combination of the first 3 vectors, so it can be deleted and the remaining three vectors form a linearly independent set that spans the same subspace as the original set of four vectors.