Let S be =

and let W be the subspace spanned by S.

=

**Is in W? If so find a linear combination of the vectors in S Which adds up to U**
So i know that i am trying to find r1, r2, r3, r4, such that

So i found the reduced row echelon form of the matrix and this is what i got

column 1 is x1 column 2 is x2 .... etc.

From this i can see that the 3rd column is free which shows that there are infinitely many solutions. So YES U is in W because the system has at least one solution.

so i paramaterized the solutions and set....

x3 = y

x1 + x3 = 2

x2 + x3 = 1

x4 = 1

so

x1 = 2-y

x2 = 1-y

x3 = y

x4 = 1

y can be any real # so... if y=1 then

r1 = 1

r2 = 0

r3 = 1

r4 = 1

This ^^ is one linear combination of the vectors in S that adds up to U

Did i do all this right? I just want to check. Thanks for looking it over and commenting on anything