MST10030 notes wiki

3d_scratchpad

Anonymous (anonymous@undisclosed.example.com) — 2015-01-26T17:29:17+00:00

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adjoint

Anonymous (anonymous@undisclosed.example.com) — 2015-03-26T10:32:11+00:00

Let $A$ be an $n\times n$ matrix. Recall that $C_{ij}$ is the $(i,j)$ cofactor of $A$. The matrix of cofactors of $A$ is the $n\times n$ matrix $C$ whose $(i,j)$ entry is $C_{ij}$. The adjoint of $A$ is the $n\times n$ matrix $J=C^T$, the transpose of the matrix of cofactors.

augmented_matrix

Anonymous (anonymous@undisclosed.example.com) — 2015-01-27T11:54:14+00:00

Given a system of linear equations: \begin{align*} a_{11}x_1+a_{12}x_2+\dots+a_{1m}x_m&=b_1\\ a_{21}x_1+a_{22}x_2+\dots+a_{2m}x_m&=b_2\\ \hphantom{a_{11}}\vdots \hphantom{x_1+a_{22}}\vdots\hphantom{x_2+\dots+{}a_{nn}} \vdots\ & \hphantom{{}={}\!} \vdots\\ a_{n1}x_1+a_{n2}x_2+\dots+a_{nm}x_m&=b_n \end{align*} its augmented matrix is \[ \begin{bmatrix} a_{11}&a_{12}&\dots &a_{1m}&b_1\\ a_{21}&a_{22}&\dots &a_{2m}&b_2\\ \vdots&\vdots& &\vdots&\vdots\\ a_{n1}&a_{n2}&\dots &a_{nm}&b_n \end{bmatrix}…

chapter_1

Anonymous (anonymous@undisclosed.example.com) — 2015-02-10T11:28:16+00:00

Chapter 1: Systems of linear equations Linear equations First example: a linear equation in two variables Consider the equation \[ 2x+5y=7.\] This is an equation in two variables, or indeterminates, $x$ and $y$. A solution of this equation is a pair of numbers $(a,b)\in \mathbb{R}^2$ so that if we replace $x$ with $a$ and replace $y$$b$$2a+5b$$7$$(3,1)$$2\times 3+5\times 1\ne 7$$(1,1)$$2\times 1+5\times 1=7$$(0,\tfrac 75)$$(0.5,1.2)$$(6,-1)$$(3.5,0)$$(-\tfrac32,2)$$\mathbb{R}^2$$\mathbb{R}^2…

chapter_2

Anonymous (anonymous@undisclosed.example.com) — 2015-04-01T12:23:21+00:00

Chapter 2: The algebra of matrices Definition An $n\times m$ matrix is a grid of numbers with $n$ rows and $m$ columns: \[ A=\begin{bmatrix}a_{11}&a_{12}&\dots&a_{1m}\\a_{21}&a_{22}&\dots&a_{2m}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\dots&a_{nm}\end{bmatrix}\] The $(i,j)$ entry of a matrix $A$ is $a_{ij}$, the number in row $i$ and column $j$ of $A$. Examples * If $B=\begin{bmatrix} 99&3&5\\7&-20&14\end{bmatrix}$, then $B$ is a $2\times 3$ matrix, and the $(1,1)$ entry of $B$ is $b_{11}=9…

chapter_3

Anonymous (anonymous@undisclosed.example.com) — 2015-04-09T14:06:31+00:00

Chapter 3: Vectors and geometry Recall that a $2\times 1$ column vector such as $\def\m#1{\begin{bmatrix}#1\end{bmatrix}}\m{4\\3}$ is a pair of numbers written in a column. We are also used to writing points in the plane $\mathbb R^2$ as a pair of numbersl; for example $(4,3)$ is the point obtained by starting from the origin, and moving $4$$3$$\vec v=\m{4\\3}$$4$$3$$\vec v$$\vec v=\m{4\\3}$$(0,0)$$(4,3)$$(-2,6)$$(2,9)$$(x,y)$$(x+4,y+3)$$(4,3)$$\m{4\\3}$$\vec v=\m{4\\3}$$\m{0\\0}$$\m{0\\0}+\m{4…

cofactor

Anonymous (anonymous@undisclosed.example.com) — 2015-03-05T10:58:36+00:00

The $(i,j)$ cofactor of an $n\times n$ matrix $A$ is $(-1)^{i+j}M_{ij}$, where $M_{ij}$ is the (i,j) minor of $A$.

column_vector

Anonymous (anonymous@undisclosed.example.com) — 2015-02-10T11:04:45+00:00

$\begin{bmatrix}3\\2\\4\\0\\-1\end{bmatrix}$ is a $5\times 1$ matrix. A matrix like this with one column is called a column matrix.

commute

Anonymous (anonymous@undisclosed.example.com) — 2015-02-17T10:37:32+00:00

We say that matrices $A$ and $B$ commute if $AB=BA$.

determinant_of_a_2x2_matrix

Anonymous (anonymous@undisclosed.example.com) — 2015-03-03T10:14:34+00:00

The number $ad-bc$ is called the determinant of the $2\times 2$ matrix $A=\mat{a&b\\c&d}$. We write $\det(A)=ad-bc$ for this number.

determinant_of_a_3x3_matrix

Anonymous (anonymous@undisclosed.example.com) — 2015-03-24T10:05:06+00:00

If $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}A=\mat{a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}}$ is a $3\times 3$ matrix, then \[\det A=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}.\] Here $C_{ij}$ are the cofactors of $A$. This formula is called the Laplace expansion of $\det A$ along the first row, since $a_{11}$, $a_{12}$ and $a_{13}$ make up the first row of $A$.

determinant_of_an_nxn_matrix

Anonymous (anonymous@undisclosed.example.com) — 2015-03-24T10:04:32+00:00

If $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}A=\mat{a_{11}&a_{12}&\dots&a_{1n}\\\vdots&&&\vdots\\a_{n1}&a_{n2}&\dots&a_{nn}}$ is an $n\times n$ matrix, then \[\det A=a_{11}C_{11}+a_{12}C_{12}+\dots+a_{1n}C_{1n}.\] Here $C_{ij}$ are the cofactors of $A$. This formula is called the Laplace expansion of $\det A$ along the first row, since $a_{11}, a_{12},\dots,a_{1n}$ make up the first row of $A$.

dot_product

Anonymous (anonymous@undisclosed.example.com) — 2017-04-05T15:04:13+00:00

Let $\def\m#1{\left[\begin{smallmatrix}#1\end{smallmatrix}\right]}\vec v=\m{v_1\\v_2\\\vdots\\v_n}$ and $\vec w=\m{w_1\\w_2\\\vdots\\w_n}$ be two vectors in $\mathbb{R}^n$. The dot product of $\vec v$ and $\vec w$ is the number $\vec v\cdot \vec w$ given by \[ \color{red}{\vec v\cdot\vec w=v_1w_1+v_2w_2+\dots+v_nw_n}.\]

elementary_operations_on_a_linear_system_for_slides

Anonymous (anonymous@undisclosed.example.com) — 2016-01-28T13:39:52+00:00

If we perform one of the following operations on a system of linear equations: * list the equations in a different order; or * multiply one of the equations by a non-zero real number; or * replace equation $j$ by “equation $j$ ${}+{}$ $c\times {}$ (equation $i$)”, where $c$$i\ne j$

elementary_operations_on_a_linear_system

Anonymous (anonymous@undisclosed.example.com) — 2016-01-24T17:59:13+00:00

elementary_row_operation_for_slides

Anonymous (anonymous@undisclosed.example.com) — 2016-01-24T18:05:18+00:00

Translate elementary operations on the linear system into operations on the rows of the augmented matrix: * change the order of the rows of the matrix; * multiply one of the rows of the matrix by a non-zero real number; * replace row $j$ by “row $j$ ${}+{}$ $c\times {}$ (row $i$)”, where $c$ is a non-zero real number and $i\ne j$. *

elementary_row_operation

Anonymous (anonymous@undisclosed.example.com) — 2015-03-26T10:12:31+00:00

Recall that when we form the augmented matrix of a linear system, each equation in the system becomes a row of the matrix. So we can translate the elementary operations on the linear system into corresponding operations on the rows of the matrix. We get three different types: * change the order of the rows of the matrix;$j$$j$${}+{}$$c\times {}$$i$$c$$i\ne j$

equal_matrices

Anonymous (anonymous@undisclosed.example.com) — 2015-02-17T10:28:03+00:00

Two matrices $A$ and $B$ are said to be equal if both of the following conditions hold: * $A$ and $B$ have the same size; and * every entry of $A$ is equal to the corresponding entry of $B$; in other words, for every $(i,j)$ so that $A$ and $B$ have an $(i,j)$ entry, we have $a_{ij}=b_{ij}$. When $A$ and $B$ are equal matrices, we write $A=B$$A\ne B$

equation

Anonymous (anonymous@undisclosed.example.com) — 2015-01-21T15:47:10+00:00

An equation consists of two parts: a left hand side, and a right hand side, with an equals sign = in between. The left hand side and the right hand side are both mathematical expressions, like $2x+5y$, or $\sqrt{17-x}$. For example $2x+5y=\sqrt{17-x}$ is an equation. See also * Equation

ero

Anonymous (anonymous@undisclosed.example.com) — 2015-01-29T10:41:51+00:00

eros

Anonymous (anonymous@undisclosed.example.com) — 2015-01-29T10:41:22+00:00

free_variable

Anonymous (anonymous@undisclosed.example.com) — 2015-02-05T12:20:47+00:00

A free variable in a linear system is a variable that is not a leading variable.

gaussian_elimination_algorithm

Anonymous (anonymous@undisclosed.example.com) — 2015-02-05T11:42:10+00:00

Aim: put any matrix into REF using EROs. Algorithm * Re-order the rows so that the leftmost leading entry in the matrix is in the top row. * Divide all of the top row by its leading entry, so that this entry becomes a $1$. * “Pivot about the leading 1”: subtract multiples of the top row from each row below so that all entries below the leading $1$$0$$1$

gaussian_elimination_remarks

Anonymous (anonymous@undisclosed.example.com) — 2016-02-11T11:18:56+00:00

We know that we can apply EROs to any augmented matrix into REF. Suppose the system has $n$ equations and $m$ variables, and let $k$ be the number of non-zero rows in REF. Also suppose the system is consistent: then the REF has no row of the form $[0~0~0~\dots~1]$. * $k\le n$, because there are only $n$$k$$k$$m$$k\le m$$$ \text{$$ is the number of free variables.} $$$k=m$$m-k=0$$k0$$m-k$$n

i_j_entry

Anonymous (anonymous@undisclosed.example.com) — 2015-02-10T11:24:26+00:00

The $(i,j)$ entry of a matrix $A$ is $a_{ij}$, the number in row $i$ and column $j$ of $A$.

identity_matrix

Anonymous (anonymous@undisclosed.example.com) — 2015-02-17T11:01:45+00:00

The $n\times n$ identity matrix is the $n\times n$ matrix $I_n$ with $1$s in every diagonal entry (that is, in the $(i,i)$ entry for every $i$ between $1$ and $n$), and $0$s in every other entry. So \[ I_n=\begin{bmatrix} 1&0&0&\dots&0\\0&1&0&\dots&0\\0&0&1&\dots&0\\\vdots & & &\ddots & \vdots\\0&0&0&\dots&1\end{bmatrix}.\]

inconsistent

Anonymous (anonymous@undisclosed.example.com) — 2015-02-05T11:58:12+00:00

A linear system with no solutions is called inconsistent. We can detect an inconsistent linear system, since whenever we apply EROs to put the augmented matrix into REF, we will get a row of the form $[0~0~0~\dots~0~*]$ where $*$ is non-zero.

indeterminate

Anonymous (anonymous@undisclosed.example.com) — 2015-01-21T10:34:12+00:00

“Indeterminate” is just another name for a variable.

invertible

Anonymous (anonymous@undisclosed.example.com) — 2016-02-24T16:13:30+00:00

An $n\times n$ matrix $A$ is invertible if there exists an $n\times n$ matrix $C$ so that \[ AC=I_n=C A.\] The matrix $C$ is called an inverse of $A$.

leading_entry

Anonymous (anonymous@undisclosed.example.com) — 2015-01-29T11:24:39+00:00

The leading entry of a non-zero row of a matrix is the leftmost entry which is not $0$. For example, the leading entry of the row $[0~0~0~6~2~0~3~1~0]$ is $6$.

leading_variable

Anonymous (anonymous@undisclosed.example.com) — 2015-02-04T11:58:46+00:00

Given an augmented matrix in REF (or RREF), each column except the last column corresponds to a variable. These come in two types: * leading variables are variables whose column contains the leading entry of some row; * free variables are all the other variables.

lecture_1_slides_test

Anonymous (anonymous@undisclosed.example.com) — 2016-01-24T16:22:58+00:00

* Consider the equation \[ 2x+5y=7.\] * This is an equation in two variables, or indeterminates, $x$ and $y$. * A solution of this equation is a pair of numbers $(a,b)\in \mathbb{R}^2$ so that if we replace $x$ with $a$ and replace $y$ with $b$, then the equation becomes true. * In other words, so that $2a+5b$ really is equal to $7$. $(3,1)$$2\times 3+5\times 1\ne 7$$(1,1)$$2\times 1+5\times 1=7$$(0,\tfrac 75)$$(0.5,1.2)$$(6,-1)$$(3.5,0)$$(-\tfrac32,2)$$\mathbb{R}^2$$\mathbb{R}^2$$2x+5y=…

lecture_1_slides

Anonymous (anonymous@undisclosed.example.com) — 2016-01-25T09:49:21+00:00

lecture_1

Anonymous (anonymous@undisclosed.example.com) — 2016-01-22T19:13:00+00:00

lecture_2_sides

Anonymous (anonymous@undisclosed.example.com) — 2016-01-24T16:44:14+00:00

If $a,b,c,d$ are any fixed numbers, then equation \[ ax+by+cz=d\] is a linear equation in 3 variables. When you draw the set of all solutions of a linear equation in 3 variables, you always get a plane in 3-dimensional space, $\mathbb{R}^3$. * $x+y+z=1$