# Week 2 - Linear Transformations and Matrices

## Opening Remarks

### Rotations in 2D

• $\alpha R_{\theta}(x) = R_{\theta}(\alpha x)$,
• $R_{\theta}(x+y) = R_{\theta}(x) + R_{\theta}(x)$

## Linear Transformations

• A linear transformation is a vector function that has the following two properties:
• Transforming a scaled vector is the same as scaling the transformed vector: $$L(\alpha x) = \alpha L(x)$$
• Transforming the sum of two vectors is the same as summing the two transformed vectors: $$L(x + y) = L(x) + L(y)$$
• Lemma: $L: \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation if and only if(iff) for all $u,v \in \mathbb{R}^n$ and $\alpha, \beta \in \mathbb{R}^n$ $$L(\alpha u + \beta v) = \alpha L(u) + \beta L(v)$$
• Lemma: Let $v_0, v_1, \ldots, v_{k-1} \in \mathbb{R}^n$ and let $L: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Then
• $L(v_0 + v_1 + \ldots + v_{k-1}) = L(v_0) + L(v_1) + \ldots + L(v_{k-1})$

## Mathematical Induction

• What is the Principle of Mathematical Induction(week induction)(数学归纳法)?
• if one can show that:
• (Base case) a property holds for $k = k_b$; and
• (Inductive step) if it holds for $k = K$, where $K \ge k_b$ , then it is also holds for $k = K +1$,
• then one can conclude that the property holds for all integers $k \ge k_b$ . Often $k_b = 0$ or $k_b = 1$.
• Example: To proof: $\displaystyle\sum_{i=0}^{n-1}{i} = n(n-1)/2$
• Base case: $n = 1$. For this case, we must show that $\displaystyle\sum_{i=0}^{i-1}{i} = 1(1-1)/2$
• $\displaystyle\sum_{i=0}^{i-1}{i} = 0 = 1(1-1)/2$
• So this proves the base case.
• Inductive step: Inductive Hypothesis (IH): Assume that the result is true for $n = k$ where $k \ge 1$: $\displaystyle\sum_{i=0}^{k-1}{i} = k(k-1)/2$
• We need to show that the result is then also true for $n=k+1$: $\displaystyle\sum_{i=0}^{(k+1)-1}{i} = (k+1)((k+1)-1)/2$
• Assume that $k \ge 1$, Then
• \begin{aligned}\sum_{i=0}^{(k+1)-1}{i} &= \sum_{i=0}^{k-1}{i} + k \\ &= k(k-1)/2 + k = k(k+1)/2 \\ &= (k+1)((k+1)-1)/2 \end{aligned}

• This proves the inductive step.

## Representing Linear Transformations as Matrices

• The Big Idea. The linear transformation L is completely described by the vectors
• $a_0 ,a_1 ,...,a_{n-1}$, where $a_j = L(e_j)$
• because for any vector x, $L(x) = \sum^{n-1}_{j=0} x_j a_j$.

### Theorem

• Let $L: \mathbb{R}^n \to \mathbb{R}^m$ be defined by $L(x) = Ax$ where $A \in \mathbb{R}^{m \times n}$. Then L is a linear transformation.
• Alternatively, A vector function $f: \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation if and only if it can be represented by an $m \times n$ matrix, which is a very special two dimensional array of numbers (elements).
• The set of all real valued $m \times n$ matrices is denoted by $\mathbb{R}^{m \times n}$

### How to check if a vector function is a linear transformation

• Check if $f(0)=0$. If it isn’t, it is not a linear transformation.
• If $f(0)=0$ then either:
• Prove it is or isn’t a linear transformation from the definition:
• Find an example where $f(\alpha x) \ne \alpha f(x)$ or $f(x + y) \ne f(x) + f(y)$. In this case the function is not a linear transformation; or
• prove that $f(\alpha x) = \alpha f(x)$ or $f(x + y) = f(x) + f(y)$ for all $\alpha, x, y$.
• or
• Compute the possible matrix A that represents it and see if $f(x)=Ax$. If it is equal, it is a linear transformation. If it is not, it is not a linear transformation.

## Some Summations will be Used in Future Weeks

• $\sum _{i=0}^{n-1} i = n ( n-1 ) / 2 \approx n^2 / 2$
• $\sum _{i=1}^{n} i = n ( n+1 ) / 2 \approx n^2 / 2$
• $\sum _{i=0}^{n-1} i^2 = (n-1) n ( 2n-1 ) / 6 \approx \frac{1}{3} n^3$

## Words

• arithmetic [ə’riθmətik, ,æriθ’metik] n. 算术，算法