> For the complete documentation index, see [llms.txt](https://lauradang.gitbook.io/notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://lauradang.gitbook.io/notes/math/math_235.md). # 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 | | ----------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | $$Col(A)$$ |

Let \vec{x} \in Col(A) and \vec{r\_1}, ... \vec{r\_m}=rows \space in \space A.
\vec{x}=c\_1\vec{r\_1}+...+c\_m\vec{r\_m}

| | $$Null(A)$$ |

Let \vec{x} \in Null(A) and \vec{r\_1}, ... \vec{r\_m}=rows \space in \space A.
\vec{0}=\begin{bmatrix}\vec{r\_1}\vec{x}\\\vdots\\\vec{r\_m}\vec{x}\end{bmatrix}

| ### 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 | |

dim(V)=n
rank(L)=dim(Range(L))

| | $$L: V\rightarrow W$$ is linear mapping | $$Ker(L)={\vec{0}}$$ | $$dim(Ker(L))=0=nullity(L)$$ | |

L: U\rightarrow V is linear mapping
M: V\rightarrow W is linear mapping

| $$Range(M \circ L)$$ |

Let \vec{x}\in Range(M\circ L).
Then there exists \vec{v}\in V where \vec{x}=(M\circ L)(\vec{v})=M(L(\vec{v}))\in Range(M).
Therefore Range(M \circ L) is a subset of Range(M)

| | **What you are trying to prove** | What to use | | -------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | $$dim V \le dim W$$ |

1. Rank nullity theorem
2. Prove that V is a subset of the W

| | $$rank(M \circ L) \le rank (M)$$ |

Prove that the Range(M \circ L) is a subset of Range(M)
- Can only use this if both ranges are in the same subspace

| | $$rank(M \circ L) \le rank (L)$$ |

Since they are not in the same subspace, we cannot use the above strategy
Instead, analyze kernels and use rank-nullity theorem.
ightarrow Prove that ker(M \circ L) \ge ker(L), then use rank-nullity theorem to do the rest

| 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$$ |

1. There exists L: V \rightarrow W that is linear mapping.
2. L is 1-1 and onto ightarrow Range(L)=W and Ker(L)={\vec{0}}

| | **What you are trying to prove** | What to use | | -------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | $$V$$ is isomorphic to $$W$$ |

1. Define a basis for V and a basis for W
2. Define mapping L: V \rightarrow W that maps the basis for V to the basis for W
e.g. V={ \vec{v\_1},...\vec{v\_2}}, W={ \vec{w\_1},...\vec{w\_2}}
L(t\_1\vec{v\_1}+...+t\_n\vec{v\_n})=t\_1\vec{w\_1}+...+t\_n\vec{w\_n}
3. Prove that the mapping is linear, 1-1, and onto.

| | $$L$$ is injective | Prove that $$ker(L)={\vec{0}}$$ |