Thank you watchmath!
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The first step is correct, they are linearly dependent and the dimension is 2. But this doesn't mean that the subspace is R^2. There are lots of subspace of dimension 2.
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Yes, you're right.
And about
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Your work basically tells you that you don't need all four vectors to generate the subspace. Two is enough. Can you choose these two vectors?
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I did chose the vectors (1,0,0,0) and (0,1,0,0) because they are the 2 columns with only a 1 and all the other elements are 0. So that was wrong?
Maye I had to consider an amplied matrix... but I thought it wasn't worth it since I thought I had found an easy answer. (that is, I could reduce the rows of the matrix that are not all 0s.)