MAT223: 13&14_Change of Basis II and Determinations

Change of Basis II §

• Change of Basis matrix - Let $\mathcal{A},\mathcal{B}$ be basis for ${\mathbb{R}}^n$. The matrix $M$ is called a change of basis matrix (which converts from $\mathcal{A}$ to $\mathcal{B}$) if for all $\vec{x}\in {\mathbb{R}}^n$

  $$M{\left[\vec{x}\right]}_{\mathcal{A}}={\left[\vec{x}\right]}_{\mathcal{B}}$$

  Notationally, $[\mathcal{B}\leftarrow \mathcal{A}]$ stands for the change of basis matrix converting from $\mathcal{A}$ to $\mathcal{B}$. Write $M=[\mathcal{B}\leftarrow \mathcal{A}]$

  

• An $n\times n$ matrix’s invertible if and only if it is a change of basis matrix

  • let $M^{-1}=[\mathcal{A}\leftarrow \mathcal{B}]$
    $$\begin{gathered} M^{-1}M=[\mathcal{A}\leftarrow \mathcal{B}][\mathcal{B}\leftarrow \mathcal{A}]=[\mathcal{A}\leftarrow \mathcal{A}]=I \\ M{M}^{-1}=[\mathcal{B}\leftarrow \mathcal{A}][\mathcal{A}\leftarrow \mathcal{B}]=[\mathcal{B}\leftarrow \mathcal{B}]=I \end{gathered}$$

Transformation and Bases §

• Linear transformation in basis - Let $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a linear transformation and let $\mathcal{B}$ be a basis for ${\mathbb{R}}^n$. The matrix for $\mathcal{T}$ wiht respect to $\mathcal{B}$, notated $[\mathcal{T}{]}_{\mathcal{B}}$ is the $n\times n$ matrix satisfying
  $$\left[\mathcal{T}\vec{x}\right]\mathcal{B}={\left[\mathcal{T}\right]}_{\mathcal{B}}{\left[\vec{x}\right]}_{\mathcal{B}}$$

In this case, say matrix $[\mathcal{T}{]}_{\mathcal{B}}$ is the representation of $\mathcal{T}$ in the $\mathcal{B}$ basis.

Similar Matrices §

• Similar Matrices - the matrices $A$ and $B$ are called similar matrices, denoted $A\sim B$, if $A$ and $B$ represent the same linear transformation but in possibly different basis. Equivalently, $A\sim B$ if there is an invertible matrix $X$ so that
  $$A=XB{X}^{-1}$$

Determinants §

• Determinant - the number (which is associated with a linear transformation with the same domain and codomain) that tracks by how much a linear transformation changes area/volume

Volumes §

• Unit $n$-cube - the unit n-cube is the $n$-dimensional cube with sides given by the standard basis vectors and lower-left corner located at the origin. That is

  $$C_n=\left\{\vec{x}\in {\mathbb{R}}^n:\vec{x}=\sum _{i=1}^n{\alpha }_i{\vec{e}}_i\text{ for some }{\alpha }_1,\dots ,{\alpha }_n\in [0,1]\right\}=[0,1{]}^n$$
  • ($C_n$ always have volume 1)

• Let $Vol(X)$ stand for the volume of the set $X$. Given a linear transformation $\mathcal{S}:{\mathbb{R}}^n\to {\mathbb{R}}^n$, define a number

  $$\text{Vol Change}(\mathcal{S})=\frac{Vol(\mathcal{S}(C_n))}{Vol(C_n)}=\frac{Vol(\mathcal{S}(C_n))}{1}=Vol(\mathcal{S}(C_n))$$

A priori, $\text{Vol Change}(\mathcal{S})$ only describes how $\mathcal{S}$ changes the volume of $C_n$. However, because $\mathcal{S}$ is a linear transformation, it actually describes how $\mathcal{S}$ changes the volume of any figure.

• Let $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a linear transformation and let $X\subseteq {\mathbb{R}}^n$ be a subset with volume $\alpha$. Then the volume of $\mathcal{T}(X)$ is $\alpha \cdot \text{Vol Change}(\mathcal{T})$

• Suppose $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a linear transformation, $X\subseteq {\mathbb{R}}^n$ is a subset, and the volume of $\mathcal{T}(X)$ is $\alpha$. Then for any $\vec{p}\in {\mathbb{R}}^n$, the volume of $\mathcal{T}(X+\left\{\vec{p}\right\})$ is $\alpha$

  $$\mathcal{T}(X+\left\{\vec{p}\right\})=\mathcal{T}(X)+\mathcal{T}(\left\{\vec{p}\right\})=\mathcal{T}(X)+\left\{\mathcal{T}(\vec{p})\right\}$$

• Fix $k$ and let $B_n$ be $C_n$ scaled to have side length $\frac{1}{k}$ and let $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a linear transformation, then

  $$\text{Vol Change}(\mathcal{T})=\frac{Vol(\mathcal{T}(B_n))}{Vol({\mathcal{B}}_n)}$$
  

  The argument now gets to

  $$\text{Vol Change}(\mathcal{T})=\frac{Vol(\mathcal{T}(C_n))}{Vol(C_n)}=\frac{K^{n}Vol(\mathcal{T}(B_n))}{k^{n}Vol({\mathcal{B}}_n)}=\frac{Vol(\mathcal{T}(B_n))}{Vol({\mathcal{B}}_n)}$$
  

The determinant §

• Determinant - the determinant of a linear transformation $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^n$, denoted $det(\mathcal{T})$ or $|\mathcal{T}|$, is the oriented volume of the image of the unit $n$-cube. The determinant of a square matrix is the determinant of its induced transformation

• Orientation Preserving Linear Transformation - Let $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a linear transformation. Say that $\mathcal{T}$ is orientation preserving if the ordered bassi $\left\{\mathcal{T}({\vec{e}}_1),\dots ,\mathcal{T}(\vec{e}n)\right\}$ is positively orientated and say that $\mathcal{T}$ is orientation reversing if ordered basis $\left\{\mathcal{T}({\vec{e}}_1),\dots ,\mathcal{T}(\vec{e}n)\right\}$ is not a basis of ${\mathbb{R}}^n$, then $\mathcal{T}$ is neither orientation preserving nor orientation reversing

  
  • Write example here

Determinants of composition §

• Let $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $\mathcal{S}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be linear transformation. Then
  $$\text{det}(\mathcal{S}\circ \mathcal{T})=\text{det}(\mathcal{S})\text{det}(\mathcal{T})$$

Determinants of matrices §

• let $A$ and $b$ be $n\times n$ matrices, then

  $$\text{det}(AB)=\text{det}(A)\text{det}(B)$$

• Volume Theorem I - For a square matrix $M$, $det(M)$ is the oriented volume of the parallelepiped (the $n$-dimensional analog of a parallelogram) given by the column vectors

• The determinants of elementary matrices are all easy to compute and determinants of the most-used typ eof elementary matrix is 1

Determinants and invariability §

• Let $A$ be an $n\times n$ matrix .$A$ is invertible if and only if $det(A)\ne 0$
  $$\text{det}(A)=\text{det}(E_1,\dots E_k)=\text{det}(E_1)\dots \text{det}(E_k)$$
  elementary matrices have non-zero determinations, so $det(A)\ne 0$

Determinants and transposes §

• Volume Theorem II - The determinant of a square matrix $A$ is equal to the oriented volume of the parallelepiped by the rows of $A$