- #1
franz32
- 133
- 0
Here's the question.
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?
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?