Linear Algebra, 2020a

254 Chapter Three. Maps Between Spaces
(b) Matrix multiplication is associative, so in computing H 1 H 2 H 3 H 4 we can expect
to get the same answer no matter where we put the parentheses. The cost in
number of multiplications, however, varies. Find the association requiring the
fewest real number multiplications to compute the matrix product of a 5×10
matrix, a 10×20 matrix, a 20×5 matrix, and a 5×1 matrix. Use the same formula
as in the prior part.
(c) (Very hard.) Find a way to multiply two 2×2 matrices using only seven
multiplications instead of the eight suggested by the prior approach.
? 3.49 [Putnam, 1990, A-5] IfA and B are square matrices of the same size such that
ABAB = 0, does it follow that BABA = 0?
3.50 [Am. Math. Mon., Dec. 1966] Demonstrate these four assertions to get an alternate
proof that column rank equals row rank.
(a) ⃗y · ⃗y = 0 iff ⃗y = ⃗0.
(b) A⃗x = ⃗0 iff A T A⃗x = ⃗0.
(c) dim(R(A)) = dim(R(A T A)).
(d) col rank(A) =col rank(A T )=row rank(A).
3.51 [Ackerson] Prove (where A is an n×n matrix and so defines a transformation of
any n-dimensional space V with respect to B, B where B is a basis) that dim(R(A)∩
N (A)) = dim(R(A)) − dim(R(A 2 )). Conclude
(a) N (A) ⊂ R(A) iff dim(N (A)) = dim(R(A)) − dim(R(A 2 ));
(b) R(A) ⊆ N (A) iff A 2 = 0;
(c) R(A) =N (A) iff A 2 = 0 and dim(N (A)) = dim(R(A)) ;
(d) dim(R(A) ∩ N (A)) = 0 iff dim(R(A)) = dim(R(A 2 )) ;
(e) (Requires the Direct Sum subsection, which is optional.) V = R(A) ⊕ N (A)
iff dim(R(A)) = dim(R(A 2 )).
IV.4
Inverses
We finish this section by considering how to represent the inverse of a linear map.
We first recall some things about inverses. Where π: R 3 → R 2 is the projection
map and ι: R 2 → R 3 is the embedding
⎛ ⎞
⎛ ⎞
x
⎜ ⎟
⎝y⎠
z
π
↦−→
(
x
y
) (
x
y
)
ι
↦−→
x
⎜ ⎟
⎝y⎠
0
then the composition π ◦ ι is the identity map π ◦ ι = id on R 2 .
⎛ ⎞
( ) x ( )
x ι ⎜ ⎟
↦−→ ⎝y⎠
↦−→
π x
y
y
0