MATH 323 Lecture 3

« previous | Tuesday, September 4, 2012 | next »

Homogeneous System

A system of equations is said to be homogeneous if all constants on the right-hand side of the equations are 0.

Homogeneous systems are always consistent usually with the trivial solution $(x_1,x_2,\dots ,x_n)=(0,0,\dots ,0)$

Theorem. An $m\times n$ homogeneous system of linear equations has a nontrivial solution iff $n>m$.

Proof. With fewer equations than unknowns, there will be at least

$n-m$

free variables. Any free variable has an infinite number of possible values, therefore the system has infinitely many solutions.

Q.E.D.

Matrix Notation

Matrices use CAPITAL LETTERS ($A$, $B$, etc.)

Entries use lowercase letters ($a$, $b$, etc.)

With $m$ rows and $n$ columns, a matrix is said to be of size $m\times n$

$A=\left[\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \dots & a_{mn}\end{array}\right]=\left(a_{ij}\right)$

A single element may be referenced with subscript row $i$, column $j$: $a_{ij}$

A row $i$ may be extracted with the following notation:

$\overrightarrow{a}(i,:)=(a_{i1},\dots ,a_{in})$

A column $j$ may be extracted with the following notation:

$\overrightarrow{a}(:,j)={\overrightarrow{a}}_j=\left[\begin{array}{c}{a}_{1j}\\ ⋮\\ a_{mj}\end{array}\right]$

Matrix constructor form:

$M_{m\times n}(\mathbb{ℝ})=\text{set (ring) of }m\times n\text{ matrices over }\mathbb{ℝ}$

Vectors

2 forms:

1. Row vector ($1\times n$): $\overrightarrow{v}=(a_{11},\dots ,a_{1n})$
2. Column vector ($n\times 1$): $\left[\begin{array}{c}{a}_{11}\\ ⋮\\ a_{n1}\end{array}\right]$

Vectors of either type constitute $m$-dimensional vector space (represented by ${\mathbb{ℝ}}^m$) in Euclidean space

If ${\overrightarrow{a}}_1,\dots ,{\overrightarrow{a}}_n$ are vectors in ${\mathbb{ℝ}}^m$ and $c_1,\dots ,c_n$ are scalars, then their sum is of the form $c_1{\overrightarrow{a}}_1+\dots +c_n{\overrightarrow{a}}_n$ and is said to be a linear combination of the vectors $a_1,\dots ,a_n$

Theorem 1.3.1. A system of equations is consistent iff $\overrightarrow{b}$ is a linear combination of column vectors ${\overrightarrow{a}}_i$ of the coefficient matrix $A$.

$x_1{\overrightarrow{a}}_1+\dots +x_n{\overrightarrow{a}}_n=\overrightarrow{b}$

Operations

Addition

Two matrices may be added only if they have the same size:

$\begin{array}{rl}A+B& =C\\ c_{ij}=a_{ij}+b_{ij}\end{array}$

Just add the corresponding cells of each matrix

Example

$\left[\begin{array}{ccc}2& -1& 3\\ -1& -2& 1\end{array}\right]+\left[\begin{array}{ccc}-1& 0& 1\\ 0& -2& -2\end{array}\right]=\left[\begin{array}{ccc}1& -1& 4\\ -1& -4& 3\end{array}\right]$

Multiplication

Matrix $A$ may multiply into matrix $B$ only if $A$ has size $m\times n$ and $B$ has size $n\times k$. The resulting matrix will have size $m\times k$

Take row $i$, multiply each of its $n$ cells by the corresponding cells of column $j$. The resulting matrix cell will be the sum of the components' products:

$c_{ij}=\sum _{t=1}^n{a}_{it}{b}_{tj}$
where $1\le i\le m$ and $1\le j\le k$

It's just a dot product!

Warning: $AB\ne BA$

Application

Consider the following equation: $2x_1-3x_2+x_3=1$

Write the coefficient matrix:

$\left[\begin{array}{ccc}2& -3& 1\end{array}\right]$

... and vector of unknowns:

$\overrightarrow{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$

Multiplying the coefficients by the unknowns yields the same equation:

$\left[\begin{array}{ccc}2& -3& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}2x_1-3x_2+x_3\end{array}\right]$

Example

<math>\begin{bmatrix}2 & -1 & 1 \\ -1 & 0 & 1\end{bmatrix} \, \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 + (-1) \cdot 0 + 1 \cdot 1 & 2 \cdot (-1) + (-1) \cdot 2 + 1 \cdot (-1) \\ (-1) \cdot 1 + 0 \cdot 0 + 1 \cdot 1 & (-1) \cdot (-1) + 0 \cdot 2 + 1 \cdot (-1) \end{bmatrix} = \begin{bmatrix} 3 & -5 \\ 0 & 0 \end{bmatrix}