$ \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}} $
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A linear transformation is determined by how it acts on a basis

Let $T:V\to W$ be a linear transformation, and $\{v_i\}_{i\in I}$ be a basis for $V$, and let $\{w_i\}_{i\in V}$ be any vectors in $W$. Then there's an unique linear transformation $T$ such that $Tv_1=w_1$ for all $i$.

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A linear transformation is determined by how it acts on a basis

Let $T:V\to W$ be a linear transformation, and $\{v_i\}_{i\in I}$ be a basis for $V$, and let $\{w_i\}_{i\in V}$ be any vectors in $W$. Then there's an unique linear transformation $T$ such that $Tv_1=w_1$ for all $i$.

Concepts

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Hypothesis

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Proof

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FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

A linear transformation is determined by how it acts on a basis

Let $T:V\to W$ be a linear transformation, and $\{v_i\}_{i\in I}$ be a basis for $V$, and let $\{w_i\}_{i\in V}$ be any vectors in $W$. Then there's an unique linear transformation $T$ such that $Tv_1=w_1$ for all $i$.

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon

A linear transformation is determined by how it acts on a basis

Let $T:V\to W$ be a linear transformation, and $\{v_i\}_{i\in I}$ be a basis for $V$, and let $\{w_i\}_{i\in V}$ be any vectors in $W$. Then there's an unique linear transformation $T$ such that $Tv_1=w_1$ for all $i$.

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon
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concepts
hypothesis
implications
proof