Find b1 in S, b2 in ortho. complement of S so that b1+b2=b=

[svalik]

1.) Find a basis for the subspace S in R4 spanned by all solutions of x1+x2+x3-x4=0

2.) Find a basis for the orthogonal complement of S

3.) Find b1 in S and b2 in the orthogonal complement of S so that b1+b2 = b = (1,1,1,1)

No idea how to solve this... Please help!!

 

[Unco]

Hi Svalik,

1.) Find a basis for the subspace S in R4 spanned by all solutions of x1+x2+x3-x4=0

Consider a similar example in R2: x1 + x2 = 0. There is one equation and two variables, so there is (two minus one = ) one parameter; let x2 = t. Then x1 = -t, and the solutions are given by x = (x1,x2) = (-t, t) = (-1,1)t. The subspace spanned by all the solutions is that spanned by (-1,1) -- remember t is a parameter -- so (-1,1) is a basis.

2.) Find a basis for the orthogonal complement of S

Your first task is to review the definition of "orthogonal complement"!