Study things that "look like"

Ex. The plane in is __not__ but is similar (plane through in

A subspace of is a subset of satisfying:

- in
- If in , then is also in (closed under addition)
- If in and is a scalar, then is in (closed under scalars)

## Example

Which are subspaces of ?

- Fails all 3 conditions ❌
- has the origin ✔️, closed under addition ✔️, closed under scalars ❌ negative is outside
- has the origin ✔️, closed under addition ❌, closed under scalars ✔️
- has the origin ❌
- has the origin ✔️, closed under addition ✔️, closed under scalars ✔️

Span

Spans ↔ Subspaces

Every span of vectors in is a subspace of

AND

Every subspace of is the span of the same vectors

If Span , say is the subspace spanned by

# Two Key Subspaces

Let be an matrix

**Column Space of**

Span of the columns of

Subspace of

**Null Space of**

Solution set to

Subspace of

## Example

Find and draw the Nul A and Col A

Col A = Span , a line in

Nul A

, ,

Nul A = Span

# To find Spanning set for...

- Nul A: Parametric vector form for solution set to
- Col A: Pivot columns will span Col A

# Basis

Let be a subspace of . A basis of is a set of vectors in that is

- linearly independent
- Span

AND

Any subspace usually has many different bases, but every basis of has the same number of vectors, the dimension of

## Example

Standard basis for

These are the unit coordinate vectors

Basis for Nul A: use parametric vector form for solution set, solve

Basis for Col A: use pivot columns of

## Example 2

Find basis for Col and Nul

Take the pivot columns in the RREF from the original matrix as Col

Nul :

## Basis Theorem

If is a subspace and dimension of

- Any linearly independent vector in is a basis for
- Any collection of vectors in that spans form a basis of

## Rank Theorem

If is matrix: rank() + nullity() =

num of pivot columns

dimension of Col

num of free variables in

dimension of Nul A