Unit 3 Continued

Review:

A is a matrix COL(A) is the span of the columns. NULL(A) is the set of solutions of $A\vec{x}=\vec{0}$

DIM(NULL(A)) is called the nullity of A. DIM(COL(A) = RANK(A). Find bases for COL(A) and NULL(A).

Example

Find bases for COL(A) and NULL(A).

This reduces to:

Basis of the column space of A:

For the Nullity, continue to RREF:

General solution:

Basis of NULL(A) =

Rank and Nullity Theorem

RANK(A) + NULLITY(A) = # of columns of A

RANK(A) = the number of columns with leading entries in the RREF

NULLITY(A) = the number of columns without leading entries

Theorem

RANK(A) = RANK($A^T$)

Equivalently

DIM(COL(A)) = DIM(COL($A^T$))

Theorem

NULL($A^{T}A$) = NULL(A)

RANK($A^{T}A$) = RANK(A)

If $B=A^T$

RANK($B^{T}B$)= RANK(B)

RANK($A{A}^T$) = RANK($A^T$) = RANK(A)

Unit 4

Linear Transformations

A transformation is a function

From ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$.

A linear transformation T is a function that satisfies:

Example:

Example

Let T be the function transformation from ${\mathbb{R}}^2$ to ${\mathbb{R}}^3$ defined by

Then:

So:

Matrix Associated to a Linear Transformation

Suppose T is a linear transformation. Define from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$.

Define the matrix $\left[\begin{array}{c}T\end{array}\right]$ whose i-th column is $T(e_i)$ where $e_i$ is the i-th column of $I_N$.

$T(\vec{v})=\left[\begin{array}{c}T\end{array}\right]\vec{v}$