Let u v w x R2 be vectors If u is a linear combination of u

Let u, v, w, x, R^2 be vectors. If u is a linear combination of u and w and v is a linear combination of w and x. show that u is a linear combination of w and x.

Solution

if u,v,w,x, belongs R^2 be vectors. if u is a linear combination of v and w and v is a linear combination of w and x.show that w is a linear combination of u and v?

If u, v, w, x R² and , R, then a vector of the form u + v is a linear combination of w and x.

Let V1 be a subspace of V and contains all the linear combinations of w1, w2, · · · , wr . Then W V1.

To prove this is a subspace we need to show that the sum of two vectors in S also belongs to S and also that any multiple of a vector in R also belongs to R. Let u, v R then we have u = (a, b, 0) and v = (c, d, 0) for some a, b, c, d R. Now

u + v = (a, b, 0) + (c, d, 0) = (a + c, b + d, 0)

which belongs to S. Also for any R

u = (a, b, 0) = ( a, b, 0) S.