Linear Algebra - 2

Theorem : Existence and Uniqueness Theorem

A linear system is consistent iff the right most column of the augmented matrix is not a pivot column. That the echelon form of the augmented matrix has no row of the form:

$$[0\cdots 0\;b]\;\; \text{b is nonzero}$$

To solve a Linear System :

1. Write the augmented matrix
2. Use the reduction algorithm to obtain an equivalent augmented matrix and decide whether is consistent
3. If it has solution, continue row reduction to get the

Section 1.3 Vector Equations

A matrix with only one column is called a column vector, or vector

Vectors in $\R^2$ : The set of all vectors with 2 entries is denoted by $\R^2$ (read “r-two”)

• sum: iff two vectors have the same size
• scalar multiple : $cu$

Geometric Descriptions

Consider a rectangular coordinate system in the plane.

We can identify a geometric point $(a,b)$ with the column vector $\begin{bmatrix} a \\ b \end{bmatrix}$

Vectors in $\R^n$:

$$\begin{bmatrix} u_1 \\ u_2 \\ \cdots \\ u_n\\ \end{bmatrix}$$

Linear Combinations

Given vectors $v_1,v_2,\cdots,v_p$ in $\R^n$ and scalars $c_1,c_2,\cdots,c_p$, the vector defined by:

$$y=c_1v_1+\cdots+c_pv_p$$

is called a linear combination of $v_1,\cdots,v_p$ with weights $c_1,\cdots,c_p$

Determine whether $b$ can be generated (or written) as a linear combination of $a_1$ and $a_2$,

Definition :

The Span $\{v_1,\cdots,v_p\}$ is called the subset of $\R^n$ spanned (or generated) by $v_1,\cdots,v_n$, which is the collection of all the ：

$$c_1v_1+c_2v_2+\cdots+c_pv_p$$

Section 1.4 Matrix Equation Ax=b

Definition

The $Ax=b$ is called a matrix equation

$A$ is an $m\times n$ matrix, and $b$ is in $R^m$, and $x$ has the same solution set of :

$$\begin{bmatrix} a_1 & a_2&\cdots &a_n&b \end{bmatrix}$$

Existence of Solutions

The equation $Ax=b$ has a solution iff $b$ is a linear combination of the columns of $A$.

This four statements are equivalent :

• For each $b$ in $\R^m$, the equation $Ax=b$ has a solution
• Each $b$ in $\R^m$ is a linear combination of the columns of $A$
• The columns of $A$ span $\R^m$
• $A$ has a pivot position in every row.

A fast way to calculate:

for $Ix=I$, $I$ is called the Identity matrix :

$$\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix}$$

Row-Vector Rule for Computing $Ax$ :

If the product $Ax$ is defined, then the $i$ th entry in $Ax$ is the sum of the products of corresponding entries of row $i$ of $A$ and from the vector $x$

Properties of Ax

$A$ is a matrix, $u,v$ are vectors in $R^n$, and $c$ is a scalar

• $A(u+v)=Au+Av$
• $A(cu)=c(Au)$