Page 185 - Matrix Analysis & Applied Linear Algebra

180 Chapter 4 Vector Spaces

4.2.10. Let p be one particular solution of a linear system Ax = b.
(a) Explain the significance of the set

p + N (A)= {p + h | h ∈ N (A)} .
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(b) If rank (A 3×3 )=1, sketch a picture of p + N (A)in .
(c) Repeat part (b) for the case when rank (A 3×3 )=2.

4.2.11. Suppose that Ax = b is a consistent system of linear equations, and
T T
let a ∈ R A . Prove that the inner product a x is constant for all
solutions to Ax = b.

4.2.12. For matrices such that the product AB is defined, explain why each of
the following statements is true.
(a) R (AB) ⊆ R (A).
(b) N (AB) ⊇ N (B).

4.2.13. Suppose that B = {b 1 , b 2 ,..., b n } is a spanning set for R (B). Prove
that A(B)= {Ab 1 , Ab 2 ,..., Ab n } is a spanning set for R (AB).