MATH 645

Axiom 1

$$u+v\in V$$

Axiom 2

$$u+v=v+u$$

Axiom 3

$$\left(u+v\right)+w=u+\left(v+w\right)$$

Axiom 4

$0\in V$ s.t $0+u=u$

Axiom 5

$-u\in V$ s.t $u+\left(-u\right)=0$

Axiom 6

$$c\cdot u\in V$$

Axiom 7

$$c\left(u+v\right)=cu+cv$$

Axiom 8

$$\left(c+d\right)u=cu+du$$

Axiom 9

$$c\cdot\left(du\right)=d\left(cu\right)$$

Axiom 10

$I\in Vs.t$ $I\cdot u=u$

What are all the addition Axioms

1. $$u+v\in V$$

2. $$u+v=v+u$$

3. $$\left(u+v\right)+w=u+\left(v+w\right)$$

4. $0\in V$ s.t $0+u=u$

5. $-u\in V$ s.t $u+\left(-u\right)=0$

What are the scalar Axioms

6. $$c\cdot u\in V$$

7. $$c\left(u+v\right)=cu+cv$$

8. $$\left(c+d\right)u=cu+du$$

9. $$c\cdot\left(du\right)=d\left(cu\right)$$

10. $I\in Vs.t$ $I\cdot U=U$

How to find the Null Space

1. Make problem into Ax = 0

2. Row reduce

3. Set all the pivots equal to the free variables (non pivots)

4. Have an X column with all the variables

5. Separate the free variables into separate columns (This is span)

6. Each column is a null space vector

Inverse of a 2×2

$$\frac{1}{ad-bc}\begin{bmatrix}d & -b\\ -c & a\end{bmatrix}$$

Diagonalization Process

1. Get Diagonal Matrix with eigenvals of A

2. Find all linearly independent eigenvectors of A

   1. If the amount of vectors is less than n for an A = n x n matrix, the matrix is not diagonalizable

3. Construct P using the eigenvectors,

4. Construct D from the eigenvalues like an Identity matrix

   1. For repeated roots, just have the repeats show twice

5. Have $A=PDP^{-1}$

6. Multiply the Matrices Left to Right

Diagonalization k Identity

When $A=PDP^{-1}$, we get$A^{k}=PD^{k}P^{-1}$