# MAT 310 : Linear Algebra

## Vector Space

### A Vector Space V over a Field F is a set with the following properties:

If X,Y, Z are in V and b,c is in F then

- X+(Y+Z) = (X+Y)+Z
- X+Y = Y+X
- There's O in V such that X+O=X
- For All v in V, There's -v such that V+ -v = O
- c*(X+Y) = c*X+c*Y
- (c+b)*X = c*X+b*X
- c*(b*X) = (c*b)*X
- 1*X=X

## Normed Vector Space

### A Normed Vector Space is a Vector Space V with a Norm. A Norm N is a function from V to F with the following properties:

If X,Y are in V and c is in F then

- N(X)≥0
- N(X)=0 iff X=0

- N(c*X) = |c|*N(X)
- N(X+Y) ≤ N(X) + N(Y)

## Inner Product Space

### An Inner Product Space is a Vector Space V with an Inner Product. An Inner Product I is a function from VxV to F with the following properties:

If X,Y, Z are in V and c is in F then

- I(X, Y) = I(Y, X)
- I(cX, Y) = cI(X, Y)
- I(X+Y, Z) = I(X, Z) + I(Y, Z)
- I(X, Y)≥0
- I(X, X)=0 iff X=0