Orthogonal decomposition

413

Let A=
[ 1 -2 -1 2]
[-1 0 3 -2 ]
[ 3 -4 -5 6]

(sorry, can't line up the columns)

I think I've done a) and b) correctly, I don't really understand c), d) and e)


a) Find RowA and Nul A

RowA={( 1, -2, -1, 2), (0, 1, -1, 0)}
Nul A= {(3, 1, 1, 0), (-2, 0, 0, 1)}

b) If b=(10 -8 28), find parametric vector form of x, the general vector in the set S={ x | Ax=b }


x= (8, -1, 0, 0) + x3 (3, 1, 1, 0) +x4 (-2, 0, 0, 1)


c) We know that (RowA) perp = Nul A. Find the orthogonal decompositon of x, the vector from b), as a sum of two vectors r ∈ RowA and n ∈ NulA.

d) Use c) to find the set S ∩ Row A, the set of elements that are in both S and RowA.

e) Suppose that A is an m x n matrix. Prove if the set S={ x | Ax=b } is non empty, then the set S ∩ RowA consists of a single vector.

Thanks.

413

nevermind, i figured it out.

ZioX

The rowspace of A has two vectors?