MATH 323 Lecture 3

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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 $(x_{1},x_{2},\ldots ,x_{n})=(0,0,\ldots ,0)$

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 $m\times n$

$A={\begin{bmatrix}a_{11}&\dots &a_{1n}\\\vdots &\ddots &\vdots \\a_{m1}&\dots &a_{mn}\end{bmatrix}}=\left(a_{ij}\right)$

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

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

${\vec {a}}(i,:)=(a_{i1},\ldots ,a_{in})$

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

${\vec {a}}(:,j)={\vec {a}}_{j}={\begin{bmatrix}a_{1j}\\\vdots \\a_{mj}\end{bmatrix}}$

Matrix constructor form:

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

Vectors

2 forms:

1. Row vector ($1\times n$): ${\vec {v}}=(a_{11},\ldots ,a_{1n})$
2. Column vector ($n\times 1$): ${\begin{bmatrix}a_{11}\\\vdots \\a_{n1}\end{bmatrix}}$

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

If ${\vec {a}}_{1},\ldots ,{\vec {a}}_{n}$ are vectors in $\mathbb {R} ^{m}$ and $c_{1},\ldots ,c_{n}$ are scalars, then their sum is of the form $c_{1}{\vec {a}}_{1}+\dots +c_{n}{\vec {a}}_{n}$ and is said to be a linear combination of the vectors $a_{1},\ldots ,a_{n}$

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

$x_{1}{\vec {a}}_{1}+\dots +x_{n}{\vec {a}}_{n}={\vec {b}}$

Operations

Addition

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

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

Just add the corresponding cells of each matrix

Example

${\begin{bmatrix}2&-1&3\\-1&-2&1\end{bmatrix}}+{\begin{bmatrix}-1&0&1\\0&-2&-2\end{bmatrix}}={\begin{bmatrix}1&-1&4\\-1&-4&3\end{bmatrix}}$

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 $1\leq i\leq m$ and $1\leq j\leq k$

It's just a dot product!

Warning: $AB\neq BA$

Application

Consider the following equation: $2x_{1}-3x_{2}+x_{3}=1$

Write the coefficient matrix:

${\begin{bmatrix}2&-3&1\end{bmatrix}}$

... and vector of unknowns:

${\vec {x}}={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}$

Multiplying the coefficients by the unknowns yields the same equation:

${\begin{bmatrix}2&-3&1\end{bmatrix}}\,{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}2x_{1}-3x_{2}+x_{3}\end{bmatrix}}$

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}