# Thread: Basis and Dimension proof

1. ## Basis and Dimension proof

Let A be a 2x2 matrix. Prove that there are real numbers a0, a1, ... a4, not all zero, such that
a4(A^4) + a3(A^3) + a2(A^2) + a1(A) + a0(I) = O

I've got no clue...I think it relates to basis/dimension since that is the part of the book this problem is in. But I don't know how to relate it to that.

2. The space of 2x2 matrices has dimension 4 (use the canonical basis), hence every basis for it has four elements. Since basis are maximal linear independent sets, it follows that every set with five elements is linearly dependent ie. there exist scalars not all zero such that the linear combination is zero

3. Just slightly different words: what you have is a linear combination of 5 vectors. Since the space of 2 by 2 matrices is of dimension 4, a set of 5 vectors cannot be independent.