$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
FullPage
definition
concepts
used in
hypothesis
results
Let $V$ and $W$ be vector spaces over $\mathbb{F}$. A transformation $A:V\to W$ is a linear transformation if
  1. $A(x+y)=Ax+Ay$ for $x, y \in\mathbb{F}^n$
  2. $A(cx)=cA(x)$ for $c\in\mathbb{F},x\in\mathbb{F}^n$

Concepts

A shorter check for linear transformation is $T(\alpha_1 x_1+\alpha_2 x_x)=\alpha_1 T x_1+\alpha_2 T x_2$.
Let $A\inM_{m, n}(\mathbb{F})$ be a transformation $A:\mathbb{F}^n\to\mathbb{F}^m$, where $A=[a_{ij}]$ and $x=\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}\in \mathbb{F}^n$ and $[x]_j=x_j$. Then $$\begin{align*} Ax:=\begin{bmatrix}a_{11}x_1+\dots + a_{1n}x_n \\ \vdots \\ a_{n1}x_1 + \dots + a_{mn}x_n\end{bmatrix} \end{align*}$$ Every linear transformation can be represented as a matrix.

Used In

Coming soon

Hypothesis

Coming soon

Results

Coming soon
Let $V$ and $W$ be vector spaces over $\mathbb{F}$. A transformation $A:V\to W$ is a linear transformation if
  1. $A(x+y)=Ax+Ay$ for $x, y \in\mathbb{F}^n$
  2. $A(cx)=cA(x)$ for $c\in\mathbb{F},x\in\mathbb{F}^n$

Concepts

A shorter check for linear transformation is $T(\alpha_1 x_1+\alpha_2 x_x)=\alpha_1 T x_1+\alpha_2 T x_2$.
Let $A\inM_{m, n}(\mathbb{F})$ be a transformation $A:\mathbb{F}^n\to\mathbb{F}^m$, where $A=[a_{ij}]$ and $x=\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}\in \mathbb{F}^n$ and $[x]_j=x_j$. Then $$\begin{align*} Ax:=\begin{bmatrix}a_{11}x_1+\dots + a_{1n}x_n \\ \vdots \\ a_{n1}x_1 + \dots + a_{mn}x_n\end{bmatrix} \end{align*}$$ Every linear transformation can be represented as a matrix.

Used In

Coming soon

Hypothesis

Coming soon

Results

Coming soon
FullPage
definition
concepts
used in
hypothesis
results
FullPage
definition
concepts
used in
hypothesis
results
Let $V$ and $W$ be vector spaces over $\mathbb{F}$. A transformation $A:V\to W$ is a linear transformation if
  1. $A(x+y)=Ax+Ay$ for $x, y \in\mathbb{F}^n$
  2. $A(cx)=cA(x)$ for $c\in\mathbb{F},x\in\mathbb{F}^n$

Concepts

A shorter check for linear transformation is $T(\alpha_1 x_1+\alpha_2 x_x)=\alpha_1 T x_1+\alpha_2 T x_2$.
Let $A\inM_{m, n}(\mathbb{F})$ be a transformation $A:\mathbb{F}^n\to\mathbb{F}^m$, where $A=[a_{ij}]$ and $x=\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}\in \mathbb{F}^n$ and $[x]_j=x_j$. Then $$\begin{align*} Ax:=\begin{bmatrix}a_{11}x_1+\dots + a_{1n}x_n \\ \vdots \\ a_{n1}x_1 + \dots + a_{mn}x_n\end{bmatrix} \end{align*}$$ Every linear transformation can be represented as a matrix.

Used In

Coming soon

Hypothesis

Coming soon

Results

Coming soon
Let $V$ and $W$ be vector spaces over $\mathbb{F}$. A transformation $A:V\to W$ is a linear transformation if
  1. $A(x+y)=Ax+Ay$ for $x, y \in\mathbb{F}^n$
  2. $A(cx)=cA(x)$ for $c\in\mathbb{F},x\in\mathbb{F}^n$

Concepts

A shorter check for linear transformation is $T(\alpha_1 x_1+\alpha_2 x_x)=\alpha_1 T x_1+\alpha_2 T x_2$.
Let $A\inM_{m, n}(\mathbb{F})$ be a transformation $A:\mathbb{F}^n\to\mathbb{F}^m$, where $A=[a_{ij}]$ and $x=\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}\in \mathbb{F}^n$ and $[x]_j=x_j$. Then $$\begin{align*} Ax:=\begin{bmatrix}a_{11}x_1+\dots + a_{1n}x_n \\ \vdots \\ a_{n1}x_1 + \dots + a_{mn}x_n\end{bmatrix} \end{align*}$$ Every linear transformation can be represented as a matrix.

Used In

Coming soon

Hypothesis

Coming soon

Results

Coming soon
FullPage
definition
concepts
used in
hypothesis
results