## Announcements:

Read Markov chains part of Section 3.7 for Wednesday. Work through recommended homework questions (and check for updates).

Midterm 2: next Thursday evening, 7-8:30 pm. Send me an e-mail message today if you have a conflict (even if you told me before midterm 1). Midterm 2 covers from Section 2.3 until the end of Chapter 3, but builds on the earlier material as well. A practice exam is available from the course home page. Last name A-Q must write in NS1, R-Z in NS7. See the missed exam section of the course web page for policies, including for illness.

Tutorials: No tutorials this week! Review in tutorials next week.

Office hour: today, 1:30-2:30, MC103B.
Help Centers: Monday-Friday 2:30-6:30 in MC 106. (But probably not Thursday or Friday this week.)

On Friday, we finished Section 3.5. That was a key section, so please study it carefully.

## New material: Section 3.6: Linear Transformations

Given an $m \times n$ matrix $A$, we can use $A$ to transform a column vector in $\R^n$ into a column vector in $\R^m$. We write: $$T_A(\vx) = A \vx \quad\text{for \vx in \R^n}$$

Example: If $$A = \bmat{rr} 0 & 1 \\ 2 & 3 \\ 4 & 5 \emat$$ then $$\kern-8ex T_A\left(\coll {-1} 2\right) = A \coll {-1} 2 = \bmat{rr} 0 & 1 \\ 2 & 3 \\ 4 & 5 \emat \coll {-1} 2 = -1 \colll 0 2 4 + 2 \colll 1 3 5 = \colll 2 4 6$$ In general (omitting parentheses), $$\kern-8ex T_A \coll x y = A \coll x y = \bmat{rr} 0 & 1 \\ 2 & 3 \\ 4 & 5 \emat \coll x y = x \colll 0 2 4 + y \colll 1 3 5 = \colll y {2x+3y} {4x + 5y}$$ Note that the matrix $A$ is visible in the last expression.

Any rule $T$ that assigns to each $\vx$ in $\R^n$ a unique vector $T(\vx)$ in $\R^m$ is called a transformation from $\R^n$ to $\R^m$ and is written $T : \R^n \to \R^m$.

For our $A$ above, we have $T_A : \R^2 \to \R^3$. $T_A$ is in fact a linear transformation.

Definition: A transformation $T : \R^n \to \R^m$ is called a linear transformation if:
1. $T(\vu + \vv) = T(\vu) + T(\vv)$ for all $\vu$ and $\vv$ in $\R^n$, and
2. $T(c \vu) = c \, T(\vu)$ for all $\vu$ in $\R^n$ and all scalars $c$.

You can check directly that our $T_A$ is linear. For example, $$\kern-8ex T_A \left( c \coll x y \right) = T_A \coll {cx} {cy} = \colll {cy} {2cx + 3cy} {4cx + 5cy} = c \colll y {2x+3y} {4x + 5y} = c \, T_A \left( \coll x y \right)$$ Check condition (1) yourself, or see Example 3.55.

In fact, every $T_A$ is linear:

Theorem 3.30: Let $A$ be an $m \times n$ matrix. Then $T_A : \R^n \to \R^m$ is a linear transformation.

Proof: Let $\vu$ and $\vv$ be vectors in $\R^n$ and let $c \in \R$. Then $$T_A(\vu + \vv) = A(\vu + \vv) = A \vu + A \vv = T_A(\vu) + T_A(\vv)$$ and $$T_A(c \vu) = A(c \vu) = c \, A \vu = c \, T_A(\vu) \qquad\Box$$

Example 3.56: Let $F : \R^2 \to \R^2$ be the transformation that sends each point to its reflection in the $x$-axis. Show that $F$ is linear. On whiteboard.

Example: Let $N : \R^2 \to \R^2$ be the transformation $$N \coll x y := \coll {xy} {x+y}$$ Is $N$ linear? On whiteboard.

It turns out that every linear transformation is a matrix transformation.

Theorem 3.31: Let $T : \R^n \to \R^m$ be a linear transformation. Then $T = T_A$, where $$A = [\, T(\ve_1) \mid T(\ve_2) \mid \cdots \mid T(\ve_n) \,]$$

Proof: We just check: \begin{aligned} T(\vx) &= T(x_1 \ve_1 + \cdots + x_n \ve_n) \\ &= x_1 T(\ve_1) + \cdots + x_n T(\ve_n) \qtext{since T is linear} \\ &= [\, T(\ve_1) \mid T(\ve_2) \mid \cdots \mid T(\ve_n) \,] \colll {x_1} {\vdots} {x_n} \\ &= A \vx = T_A(\vx) \end{aligned}

The matrix $A$ is called the standard matrix of $T$ and is written $[T]$.

Example 3.58: Let $R_\theta : \R^2 \to \R^2$ be rotation by an angle $\theta$ counterclockwise about the origin. Show that $R_\theta$ is linear and find its standard matrix.

Solution: A geometric argument shows that $R_\theta$ is linear. On whiteboard.

To find the standard matrix, we note that $$R_\theta \coll 1 0 = \coll {\cos \theta} {\sin \theta} \qqtext{and} R_\theta \coll 0 1 = \coll {-\sin \theta} {\cos \theta}$$ Therefore, the standard matrix of $R_\theta$ is $\bmat{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \emat$.

Now that we know the matrix, we can compute rotations of arbitrary vectors. For example, to rotate the point $(2, -1)$ by $60^\circ$: \begin{aligned} R_{60} \coll 2 {-1} &= \bmat{rr} \cos 60^\circ & -\sin 60^\circ \\ \sin 60^\circ & \cos 60^\circ \emat \coll 2 {-1} \\ &= \bmat{rr} 1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \emat \coll 2 {-1} = \coll {(2+\sqrt{3})/2} {(2 \sqrt{3}-1)/2} \end{aligned}

Rotations will be one of our main examples.

### New linear transformations from old

If $T : \R^m \to \R^\red{n}$ and $S : \R^\red{n} \to \R^p$, then $S(T(\vx))$ makes sense for $\vx$ in $\R^m$. The composition of $S$ and $T$ is the transformation $S \circ T : \R^m \to \R^p$ defined by $$(S \circ T)(\vx) = S(T(\vx)) .$$ If $S$ and $T$ are linear, it is easy to check that this new transformation $S \circ T$ is automatically linear. For example, \kern-8ex \begin{aligned} (S \circ T)(\vu + \vv) &= S(T(\vu + \vv)) = S(T(\vu) + T(\vv)) \\ &= S(T(\vu)) + S(T(\vv)) = (S \circ T)(\vu) + (S \circ T)(\vv) . \end{aligned} It's because of this that matrix multiplication is defined how it is. Notice also that the condition on the sizes of matrices in a product matches the requirement that $S$ and $T$ be composable.

Example 3.61: Find the standard matrix of the transformation that rotates $90^\circ$ counterclockwise and then reflects in the $x$-axis. How do $F \circ R$ and $R \circ F$ compare? On whiteboard.

Example: It is geometrically clear that $R_\theta \circ R_\phi = R_{\theta+\phi}$. This implies some trigonometric identities. On whiteboard.

.