Day 7 Outline

2.1 cont'd

Matrix multiplication is non-commutative

What shapes allow us to commute a product?

Prove non-commutativity by example!

Transpose of product?

A5by3.png B3by4.png

(AB)T = ?

A5by3T.png B3by4T.png

B3by4T.png A5by3T.png

2.2 Matrix inverse

Reminder: definitions

onto = fills up entire codomain

1-1 (aka 1-to-1) = "non-scrunching"


linear transformation R3 → R4 can't be onto: there aren't enough columns to span the codomain.


(Though a nonlinear transformation could be: example.)

linear transformation R4 → R3 has to "scrunch" (can't be 1-to-1).


Identity transformation

Def: A transformation Rn → Rn such that T(x) = x for all x is called the identity transformation (from Rn → Rn).

What is its matrix, In? (Where does T send the "ei" vectors?)

Undoing a transformation

Suppose A is the matrix of a transformation Rn → Rm

Under what circumstances could the transformation be "undone"?

In other words, when could we find a transformation Rm → Rn with matrix B such that BA gets us back to where we started?

What about a transformation with a wide matrix?

4by3.png A3by4.png

Nope: scrunching can't be undone.

What about a transformation with a tall matrix?

3by4.png A4by3.png

Maybe ...

Could we ever have an undoing matrix in both directions?

3by3.png A3by3.png

A3by3.png 3by3.png

For a square matrix, if BA = I, then AB = I also.

a matrix B with these properties is called an inverse of A, written A − 1

Q: When does a square matrix have an inverse? When it doesn't scrunch.

2x2 inverse

When does a 2x2 matrix not scrunch?

Determinant of 2x2.

Formula for 2x2 inverse. Demonstrate by brute force.

Invertible matrices

When A has an inverse, it is called invertible, or non-singular.

Thm 5:

If A is an invertible nxn matrix, then for any b, Ax=b has a unique solution, and it's A − 1b.

Thm 6:

a. If A invertible, A − 1 is invertible too and (A − 1) − 1 = A

b. If A and B are invertible nxn matrices, then AB ....

c. If A is invertible, so is AT and (AT) − 1 =  ...

Algorithm for finding A − 1

Stacking "Ax=B"s

Augment A with identity and RRE ...

2.3 Characterizations of invertible