linear-guest

G.6 Matrices 389
Hint for Review Question 4
This problem just amounts to remembering that the dot product of x = (x 1 , x 2 , . . . , x n )
and y = (y 1 , y 2 , . . . , y n ) is
x 1 y 1 + x 2 y 2 + · · · + x n y n .
Then try multiplying the above row vector times y T
Hint for Review Question 5
and compare.
The majority of the problem comes down to showing that matrices are right
distributive. Let M k is all n × k matrices for any n, and define the map
f R : M k → M m by f R (M) = MR where R is some k × m matrix. It should be
clear that f R (α · M) = (αM)R = α(MR) = αf R (M) for any scalar α. Now all
that needs to be proved is that
f R (M + N) = (M + N)R = MR + NR = f R (M) + f R (N),
and you can show this by looking at each entry.
We can actually generalize the concept of this problem. Let V be some
vector space and M be some collection of matrices, and we say that M is a
left-action on V if
(M · N) ◦ v = M ◦ (N ◦ v)
for all M, N ∈ N and v ∈ V where · denoted multiplication in M (i.e. standard
matrix multiplication) and ◦ denotes the matrix is a linear map on a vector
(i.e. M(v)). There is a corresponding notion of a right action where
v ◦ (M · N) = (v ◦ M) ◦ N
where we treat v ◦ M as M(v) as before, and note the order in which the
matrices are applied. People will often omit the left or right because they
are essentially the same, and just say that M acts on V .
Hint for Review Question 8
This is a hint for computing exponents of matrices. So what is e A if A is a
matrix? We remember that the Taylor series for
So as matrices we can think about
e x =
e A =
∞∑
n=0
∞∑
n=0
x n
n! .
A n
n! . 389