**Vector Space Axioms:**- A nonempty set
*V*of vectors, with addition and scalar multiplication defined, satisifies the following axioms for all , and scalars*c*,*d*- 1.
- 2.
**u**+**v**=**v**+**u**- 3.
- (
**u**+**v**) +**w**=**u**+ (**v**+**w**) - 4.
- There exists
with
**u**+**0**=**u**, for . - 5.
- For each there exists a vector with
- 6.
- 7.
- 8.
- 9.
- 10.
- 1
**u**=**u**

- Axioms also imply
**0**is unique; c**0**=**0**; -**u**= (-1)**u**

- A nonempty set
**Subspaces:**- A subspace
*H*of*V*is a vector space with .- Properties:
- 1.
- 2.
- 3.
**0**for*V*is same**0**for*H*

- Subpsace spanned by a set: if

then is a subspace of*V*.

- Properties:

**Null Space of***A*:- .
*Nul*(*A*) is a subspace of*R*^{n}.- Explicit description of
*Nul*(*A*):

Find a spanning set by the determining parametric solutions to .

**Column Space of***A*:- .
*Col*(*A*) is a subspace of*R*^{m}.-
*Col*(*A*) =*R*^{m}iff consistent for all .

**Kernel and Range of Linear Transformation***T*:- Suppose
for linear spaces
*V*and*W*.- The
**kernel**of . - The
**range**of .

- The

2000-07-06