Find a basis of a subspace of R^4

[greendays]

Find a basis of the given span{[2,1,0,-1],[-1,1,1,1],[2,7,4,5]}

So I got the RREF, and found the basis to be two rows of the RREF, which are [1,0,-1/3,0] and [0,1,2/3,1], but the answer is [2,1,0,-1],[-1,1,1,1]

Where did I do wrong?

 

[Robert1986]

If the first vector of the two answer vectors is supposed to be [2,1,0,-1] then you got the same answer as the book - only you are spanning the space with different vectors.

 

[jbunniii]

Find a basis of the given span{[2,1,0,-1],[-1,1,1,1],[2,7,4,5]}

So I got the RREF, and found the basis to be two rows of the RREF, which are [1,0,-1/3,0] and [0,1,2/3,1], but the answer is [2,1,0,-1],[-1,1,1,1]

Where did I do wrong?

As Robert1986 indicated, the solution is not unique, so you can have a correct answer that doesn't match the book.

However, your answer is incorrect. You didn't show your work, so I can't say what you did wrong. But your vectors do not span the same subspace as the given vectors. There is no linear combination of your two vectors that will give you [2, 1, 0, -1].

If

a[1, 0, -1/3, 0] + b[0, 1, 2/3, 1] = [2, 1, 0, -1]

then equality of the first coordinate forces a = 2, and equality of the second coordinate forces b = 1. But then you don't get equality in the fourth coordinate.

 

[Robert1986]

As Robert1986 indicated, the solution is not unique, so you can have a correct answer that doesn't match the book.

However, your answer is incorrect. You didn't show your work, so I can't say what you did wrong. But your vectors do not span the same subspace as the given vectors. There is no linear combination of your two vectors that will give you [2, 1, 0, -1].

If

a[1, 0, -1/3, 0] + b[0, 1, 2/3, 1] = [2, 1, 0, -1]

then equality of the first coordinate forces a = 2, and equality of the second coordinate forces b = 1. But then you don't get equality in the fourth coordinate.

Wow - I can't add!

 

[vela]

I find that the three vectors are linearly independent.