- #1

Supernova123

- 12

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## Homework Statement

How would one determine if a vector space is a subspace of another one? I think that the basis vectors of the subspace should be able to be formed from a linear combination of the basis vectors of the vector space.

However, that doesn't seem to be true for this question: Let matrix A consist of column vectors (1,2,-3), (-4,-4,4) and (6,2,-8) with eigenvalues -2 and -5. I found e1=(2,3,1) and e2=(1,0,-1).

The linear space spanned by e1 and e2 is denoted by V. Prove that, for any vector x belonging to V, the vector Ax also belongs to V. I tried forming any of the column vectors in matrix A through a linear combination of e1 and e2 but that seems to fail. Instead, the answer is this: Let x=ae1+be2. Ax=A(ae1+be2)=aAe1+bAe2=-2ae1-5ae2. I understand this but why doesn't my method work?