# Math 235

## General Proofs Help Table

| **What you are trying to prove** | What to use when specific theorems fail |
| -------------------------------- | --------------------------------------- |
| $$A\le B$$                       | Prove A is a subset of B                |
|                                  |                                         |
|                                  |                                         |

## 7.1 Fundamental Subspaces

| Concepts    | Representations                                                                                                                                                                                                                                      |
| ----------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $$Col(A)$$  | <p>Let <span class="math">\vec{x} \in Col(A)</span> and <span class="math">\vec{r_1}, ... \vec{r_m}=rows \space in \space A</span>.<br><span class="math">\vec{x}=c_1\vec{r_1}+...+c_m\vec{r_m}</span></p>                                           |
| $$Null(A)$$ | <p>Let <span class="math">\vec{x} \in Null(A)</span> and <span class="math">\vec{r_1}, ... \vec{r_m}=rows \space in \space A</span>.<br><span class="math">\vec{0}=\begin{bmatrix}\vec{r_1}\vec{x}\\\vdots\\\vec{r_m}\vec{x}\end{bmatrix}</span></p> |

### Extensions from Theorems

**Theorem 7.1.5**: $$rank(A)=rank(A^T)$$

$$ightarrow Col(A)=Row(A^T), Row(A)=Col(A^T)$$

## 8.2 Linear Mappings

### Inventing/Finding linear mappings

Common $$V$$ and $$W$$ vector spaces to use:

| Vector Space    | Dimension                                    |
| --------------- | -------------------------------------------- |
| $$P_2(\R)$$     | # of $$x^n$$ terms e.g. $$dim(a+bx+cx^2)=3$$ |
| $$M_{2x2}(\R)$$ | 4                                            |
| $$^n$$          | n                                            |

Practice: pg. 224 q3,4

### Proofs Help

| Definitions                                                                                                                           | Assumptions/Information                           | Interpretations/Applicable Theorems                                                                                                                                                                                                                                                                                           |
| ------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $$L: V\rightarrow W$$ is linear mapping                                                                                               | $$\{L(\vec{v_1}), ...L(\vec{v_k})\}$$ spans $$W$$ | $$Range(L)=W$$, $$dim(Range(L))=k=rank(L)$$                                                                                                                                                                                                                                                                                   |
| $$L: V\rightarrow W$$ is linear mapping                                                                                               |                                                   | <p><span class="math">dim(V)=n</span><br><span class="math">rank(L)=dim(Range(L))</span></p>                                                                                                                                                                                                                                  |
| $$L: V\rightarrow W$$ is linear mapping                                                                                               | $$Ker(L)=\{\vec{0}\}$$                            | $$dim(Ker(L))=0=nullity(L)$$                                                                                                                                                                                                                                                                                                  |
| <p><span class="math">L: U\rightarrow V</span> is linear mapping<br><span class="math">M: V\rightarrow W</span> is linear mapping</p> | $$Range(M \circ L)$$                              | <p>Let <span class="math">\vec{x}\in Range(M\circ L)</span>.<br>Then there exists <span class="math">\vec{v}\in V</span> where <span class="math">\vec{x}=(M\circ L)(\vec{v})=M(L(\vec{v}))\in Range(M)</span>.<br>Therefore <span class="math">Range(M \circ L)</span> is a subset of <span class="math">Range(M)</span></p> |

| **What you are trying to prove** | What to use                                                                                                                                                                                                                                                                                    |
| -------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $$dim V \le dim W$$              | <p>1. Rank nullity theorem<br>2. Prove that V is a subset of the W</p>                                                                                                                                                                                                                         |
| $$rank(M \circ L) \le rank (M)$$ | <p>Prove that the <span class="math">Range(M \circ L)</span> is a subset of <span class="math">Range(M)</span><br>- Can only use this if both ranges are in the same subspace</p>                                                                                                              |
| $$rank(M \circ L) \le rank (L)$$ | <p>Since they are not in the same subspace, we cannot use the above strategy<br>Instead, analyze kernels and use rank-nullity theorem.<br><span class="math">ightarrow</span> Prove that <span class="math">ker(M \circ L) \ge ker(L)</span>, then use rank-nullity theorem to do the rest</p> |

When you are given information about **rank**, try to turn it into information about **nullity** instead since you can generally do more with kernals then range.

## 8.3 Matrix of a Linear Mapping

**Solving for** $$[\vec{x}_B]$$:

* $$\vec{x}=c_1\vec{B_1}+c_2\vec{B_2}...$$ Solve for $$c_1, c_2, ...$$
  * If $$\vec{x}$$ is a polynomial, collect like terms on the RHS if $$\vec{x}$$ is in the form $$a+b\vec{x}+...$$ This way we can do coefficient equality in order to solve for the coefficients (pg. 226 example)

**Solving for** $$[L(\vec{v})]_C$$

* $$L(\vec{v})=c_1C_1+...$$ Solve for $$c_1, ...$$

## 8.4 Isomorphisms

| Assumptions/Information      | Interpretations/Applicable Theorems                                                                                                                                                                                                           |
| ---------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $$V$$ is isomorphic to $$W$$ | <p>1. There exists <span class="math">L: V \rightarrow W</span> that is linear mapping.<br>2. L is 1-1 and onto <span class="math">ightarrow</span> <span class="math">Range(L)=W</span> and <span class="math">Ker(L)=\{\vec{0}\}</span></p> |

| **What you are trying to prove** | What to use                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $$V$$ is isomorphic to $$W$$     | <p>1. Define a basis for <span class="math">V</span> and a basis for <span class="math">W</span><br>2. Define mapping <span class="math">L: V \rightarrow W</span> that maps the basis for <span class="math">V</span> to the basis for <span class="math">W</span><br>e.g. <span class="math">V=\{ \vec{v_1},...\vec{v_2}\}</span>, <span class="math">W=\{ \vec{w_1},...\vec{w_2}\}</span><br><span class="math">L(t_1\vec{v_1}+...+t_n\vec{v_n})=t_1\vec{w_1}+...+t_n\vec{w_n}</span><br>3. Prove that the mapping is linear, 1-1, and onto.</p> |
| $$L$$ is injective               | Prove that $$ker(L)=\{\vec{0}\}$$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
