Question

Let U be a supspace of R^4 such that U={(a,b,c,d): b+c+d=0}. Finds its basis and dimension.

Answer 1

dimension 3

Answer 2

how?

Answer 3

(1 1 -1 0) (1 0 1 -1) (0 1 0 -1)

Answer 4

I know that first I need to determine a basis, which will be one that consists of a seat of vectors which spans V and is also linearly independent. And since every basis MUST consist of the same number of vectors, I will get the dimension as well.

One basis according to me is: {(0,1,-2,1),(1,0,0,0)}

So... the dimension should be 2?

Answer 5

no

Answer 6

Why? :(

for any two constants c1 and c2:

c1(0,1,-2,1) + c2(1,0,0,0)= (0,0,0,0)

is only trivially true for c1=c2=0, so they are linearly independent. They also span U, similarly (you just replace (0,0,0,0) by (p, q, r, s) to check, where p, q, r and s are arbitrary elements... and you see that you can get them for different values of c1 and c2).

Please tell me where I'm going wrong.

Answer 7

it's true that they are linearly independant but they don't engender U