kevinlui's site

October 9

Announcements

• Section 2.1, 2.2 due this Thursday

• Section 2.3, 3.1 due next Thursday

• Midterm next Wednesday in class

  • 1.1 - 3.1 (maybe 3.2)

• Worksheet 1 solutions has been posted

• Worksheet 2 has been posted, due this Friday

• Wednesday office hours to be held in classroom?

Homogenous Systems

Let $A$ be a matrix. Then $A(x+y)=Ax+Ax$ and $A(x-y)=Ax-Ay$.

Example: Find a general solution for the linear system ParseError: KaTeX parse error: {align} can be used only in display mode. Using row reduction, we see that a general solution is of the form $x=(1,0,-5,0)+s_1(3,1,0,0)+s_2(-2,0,4,1)$.

The solution to the homogenous system is $x=s_1(3,1,0,0)+s_2(-2,0,4,1)$.

Let $x_p$ be a particular solution $Ax=b$. Then solutions have the form $x_g=x_p+x_h$, where $x_p$ is a particular solution and $x_h$ is the general solution to the homogenous equations.

Linear Indepedence and Span

Theorem: Let $A=[a_i]$ and $b$ be a vector in $\mathbb{R}^n$. Then the following are equivalent (if one is true then they are all true, if one is false then they are all false).

• The set $\{a_1,\ldots,a_m\}$ are linearly independent.

• The vector equation $x_1a_1+x_2a_2+\ldots+x_ma_m=b$ has at most one solution.

• The linear system $[a_1\;a_2\;\ldots\;a_m | b]$ has at most one solution.

• The equation $Ax=b$ has at most 1 solution.

Example: Consider the vectors $a_1=(1,7,-2)$, $a_2=(3,0,1)$, and $a_3=(5,2,6)$. Set $A=[a_i]$. Show that the columns of $A$ are linearly independent and that $Ax=b$ has a unique solution for every $b$ in $\mathbb{R}^3$.

Example: Let $u_1=(1,-1,2), u_2=(2,-1,2), u_3=(-2,5,-10), u_4=(3,-4,8)$. The associated matrix has reduced echelon form: $$\begin{bmatrix} 1 & 2 & -2 & 3 \\ 0 & 1 & 3 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Is $\{u_1,\ldots,u_4\}$ linearly independent? Can we write $u_1$ as a linear combination of $u_2,\ldots,u_4$?

If a set of vectors is not linearly indepedent, can every vector be written as a linear combination of the other vectors? In other words, is every vector in the span of the other vectors?

Section 3.1 Linear Transformations

We can write linear equations as $Ax=b$. We can think of it as $A$ sending $x$ to $b$.

Definition: A function $T:\mathbb{R}^m \to \mathbb{R}^n$ is a linear transformation if for all vectors $u,v\in \mathbb{R}^m$ and all scalars $r$, we have

• T(u+v) = T(u) + T(v)

• T(ru) = rT(u).

Examples:

• What are some examples of functions that aren't linear transforms? quadratic, ax+b

• Consider the function given by $T(x_1, x_2) = (3x_1-x_2, 2x_1+5x_2)$. What is $T(1,2)$? Show that this is a linear transformation. Is it associated to a matrix?

• Projections are linear transforms.

• Let $A$ be some matrix. Then $T(x)=Ax$ is a linear transform. Make up some example in class.

A matrix, $A$, is said to be an $n\times m$ matrix if it has $n$ rows and $m$ columns. If $m=n$, then $A$ is a sqaure matrix.

Theorem: Let $A$ be an $n\times m$ matrix, and define $T(x)=Ax$. Then $T:\mathbb{R}^m \to \mathbb{R}^n$ is a linear transform. Moreover, all linear transform are of this form.

Example: Consider the linear transform with matrix $$A= \begin{bmatrix} 1 & -2 & 4 \\ 3 & 0 & 5 \end{bmatrix}.$$ Is $(3,4)$ in the range of $A$?

Theorem: Let $A=[a_1\; a_2\; \ldots\; a_m$ be a $n\times m$ matrix, and let $T:\mathbb{R}^m\to\mathbb{R}^n$ with $T(x) = Ax$ be a linear transformation. Then

• A vector $w$ is in the range of $T$ if and only if $Ax=w$ is a consistent linear system.

• The range of $T$ is the span of the columns (this is also called the column space).

If time, talk about 1-1 and onto