9/14
distribution of matrix multiplication over addition:
Why is A(X + Y) = AX + AY, as claimed last time?
It's a general result, not having anything to do with column vectors:
For any matrices with the right dimensions,
A(B + C) = AB + AC
proof:
[A(B
9/2
linear combinations, spanning, linear independence, bases, and dimension:
Let v_1 . v_n be vectors in a vector space V.
A linear combination of v_1 . v_n is anything of form
Sumcfw_i = 1 to n (a_i)(v_i)
for a_1 . a_n being any scalars.
spanning:
9/23
finding Im(A):
We looked at the system
x + 2y + 3z - w = 0
x+y+z+w=0
2x + 3y + 4z = 0
and found it had the solution set
judging from the coefficient matrix
A=
12
3 -1
1
1
1
2
1
3
4
0
having the reduced form
A_R =
1 0 -1 3
0
2
-2
0
1
0
0
0
Ker
examples of finding solution sets:
homogeneous system
Suppose we have
x+y+z+w=0
x-y-z+w=0
x+w=0
Find the solution set.
We can represent this system as
AX = 0
where
A=
1
11
1
1
-1
1
1
-1
0
0
1
We change to reduced row-echelon,
A_R =
1001
0
1
0
0
0
9/28
transpose:
If A is an (m x n) matrix, then the transpose of A, or A^t, is an (n x m) matrix
defined by
[A^t]^-_j = [A]^j_i
properties:
For any matrix A and scalar c,
(cA)^t = cA^t (easy)
If A and B are both (m x n), then
(A + B)^t = A^t + B^t
8/29
outline of course:
1/3 - 1/2 on linear algebra
1/3 - 1/2 on vector calculus
2-4 weeks on Fourier series
if time, application of Fourier series to partial differential equations
matrices
m x n ("m by n") matrix has m rows, n columns
Typical example
9/16
examples with specific matrix:
Let's look at the system (*):
2x + 4y + 3z + 2w = 0
x + y + z + 2w = 1
-x - 2y - 3z + 2w = 1
This has the coefficient matrix
A=
24
1
1
3
2
1
2
-1 -2 -3
2
and the augmented matrix (augmented by adding in the right-h
9/7
things from the future:
Given a matrix A of dimension m x n, we can define two vector spaces:
K = Ker(A) = the kernel of A
= cfw_v | Av = 0
I = Im(A) = the image of A
= cfw_Av | v is in R^n
Big Fact:
dim(K) + dim(I) = # columns
If A is thought
9/12
reformulating a system of linear equations as a matrix equation:
Suppose we have a system of m simultaneous linear equations in n real variables:
system (*):
(a^1_1)x_1 + . + (a^n_1)x_n = c_1
.
(a^1_m)x_1 + . + (a^n_m)x_n = c_m
We can replace th
9/19
proving more results:
Let A_R be the reduced row-echelon form of A.
Then dim(col-sp(A_R) = dim(col-sp(A).
(Note: This does not say the column-space remains unchanged, but only its
dimension.)
Proof:
Let the columns of A (m x n) be
cfw_Col^1, .,