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A theorem says that any linearly independent subset S of vector space V can be extended to a basis for V.

1. S = {(1, 1, 0, 0) ; (1, 0, 1, 0)}, V = R^4.

In this part, the two vectors are linearly independent by verification. Then making a basis T with dimension of 4. I arrived at an answer of

T = { (1, 1, 0, 0) ; (1, 0, 1, 0); (1, 0, 0, 0); (0, 0, 0,1)}

Am I correct?

2. S = { t^3 - t + 1; t^3 + 2} , V = P(sub3)

Again, these are linearly independent by verification.

I used the natural basis { t^3, t^2, t, 1}.

And then, since the dimension is 4, here's my answer.

T = {t^3 -t +1; t^3 + 2; t^3; t^2}

But my book has a different answer....

instead of t^3 and t^2, it was t and 1 respectively. Are both

of these correct?

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# Homework Help: Can I show my solution? Linear Algebra

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