CLU Linear Algebra Ch 1 Theorems

Theorem 1

Each matrix is row equivalent to one and only one reduced echelon matrix.

Theorem 2

A linear system is consistent if and only if the right most column of the augmented matrix is not a pivot. Furthermore, if a linear system is consistent then the solution set is unique if there are no free variables and infinite if there is at least one free variable.

Theorem 3

If $A$ is a $m \times n$ matrix with columns $\bar{a}_1, \bar{a}_2, ... ,\bar{a}_n$ and if $\bar{b}$ is in $\mathbb{R}^m$, the matrix equation

(1)

\begin{align} A \bar{x} = \bar{b} \end{align}

has the same solution set as the vector equations

(2)

\begin{align} x_1\bar{a}_1 + x_2\bar{a}_2+ ... + x_n \bar{a}_n = \bar{b} \end{align}

which also has the same solution set as the augmented matrix

(3)

\begin{align} [\bar{a}_1 \text{ } \bar{a}_2 ... \bar{a}_n \text{ } \bar{b} ] \end{align}

Theorem 4

Let $A$ be an $m \times n$ matrix. Then the following statements are logically equivalent. That is, for a particular $A$, either they are all true statements or they are all false.
a. For each b in $\mathbb{R}^m$, the equation $Ax=b$ has a solution.
b. Each b in $\mathbb{R}^m$ is a linear combination of the columns of $A$.
c. The columns of $A$ span $\mathbb{R}^m$
d. $A$ has a pivot position in every row.

Theorem 5

If $A$ is an $m \times n$ matrix, $u$ and $v$ are vectors in $\mathbb{R}^n$, and c is a scalar, then:
a. $A(u+v)=Au+Av$;
b. $A(cu)=c(Au)$

Theorem 6

Suppose the equation $Ax=b$ is consistent for some given $b$, and let $p$ be a solution. Then the solution set of $Ax=b$ is the set of all vectors of the form $w=p+v_h$, where $v_h$ is any solution of the homogenous equation $Ax=0$

Theorem 7

A set containing more than 2 vectors is linearly dependent if and only if at least one vector is a linear combination of the others

Theorem 8

If a set contains more vectors than there are entries in each vector then the set is linearly dependent

Theorem 9

Every set of vectors containing $\vec{0}$ is linearly dependent

Matrix Transformations

1. Every Matrix Transformation is Linear
2. T is a linear transformation if and only if $T(c\vec{u} + d\vec{v}) = cT(\vec{u}) + dT(\vec{v}) \text{ and } T(\vec{0}) = \vec{0}$