Notes on linear algebra

NOTES ON LINEAR ALGEBRA


CONTENTS:
[4] MATRIX ADDITION
[5] MATRIX NOTATION
[6] TRANSPOSE
[7] SYMMETRIC MATRICES
[8] BASIC FACTS ABOUT MATRICES
[4] MATRIX ADDITION
Let A and B be two matrices. When can we add them, and what is the answer? We define matrix addition by adding componentwise. For example:

(1 2) ₊ (5 7) ₌ (6 9)

(3 4) (2 0) (5 4)

Or

(1 2 5) ₊ (5 7 1) ₌ (6 9 6)
(3 4 0) (2 0 8) (5 4 8)

Of course, we’ve yet to give any motivation as to why one would want to define matrix addition by the above. Remember how we introduced matrices as maps from one space to another. For example, consider the matrix

(1 2 5)
(3 4 0)

It has 2 rows and 3 columns. It acts on vectors with three components, and returns something with 2 components. For example:

(1 2 5) (3) (1*3 + 2*2 + 5*1) ( 7)

(3 4 0) (2) = (3*3 + 4*2 + 0*1) = (17)
(1)

So, if we have two matrices A and B acting on the same vector, we can now see why they should have the same number of rows and columns. They should have the same number of

columns because they both act on the same vector. They should have the same number of
rows because they should each take that vector to the same space.

For example, here’s an example of what can go wrong when we try to add two matrices of different sizes.

Consider

(1 3 2) (3) (1*3 + 3*2 + 2*1) (11)
(2 4 1) (2) = (2*3 + 4*2 + 1*1) = (15)
(4 5 1) (1) (4*3 + 5*2 + 1*1) (23)
Then

(1 2 5) (3) (1 3 2) (3) ( 7) (11)

(3 4 0) (2) + (2 4 1) (2) = (17) + (15)
(1) (4 5 1) (1) (23)

And we have trouble, as the two vectors are different sizes. One lives in the 2dimensional plane, one lives in 3space. There is no way we can write down one matrix to represent the action of the two matrices.