$ \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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proof
Let $T:V\to V$ be a linear transformation. The following are equivalent.
  1. $\lambda$ is an eigenvector for $T$
  2. $E_{\lambda}=\text{ker}(T-\lambda I)\neq 0$
  3. $T-\lambda I$ is not invertible
  4. $\text{det}(T-\lambda I)=0$

Concepts

Since $T-\lambda I$ connutes with $T$ and

If

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Only If

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Proof

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Let $T:V\to V$ be a linear transformation. The following are equivalent.
  1. $\lambda$ is an eigenvector for $T$
  2. $E_{\lambda}=\text{ker}(T-\lambda I)\neq 0$
  3. $T-\lambda I$ is not invertible
  4. $\text{det}(T-\lambda I)=0$

Concepts

Since $T-\lambda I$ connutes with $T$ and

If

Coming soon

Only If

Coming soon

Proof

Coming soon
FullPage
result
concepts
If
Only If
proof
FullPage
result
Concepts
If
Only If
proof
Let $T:V\to V$ be a linear transformation. The following are equivalent.
  1. $\lambda$ is an eigenvector for $T$
  2. $E_{\lambda}=\text{ker}(T-\lambda I)\neq 0$
  3. $T-\lambda I$ is not invertible
  4. $\text{det}(T-\lambda I)=0$

Concepts

Since $T-\lambda I$ connutes with $T$ and

If

Coming soon

Only If

Coming soon

Proof

Coming soon
Let $T:V\to V$ be a linear transformation. The following are equivalent.
  1. $\lambda$ is an eigenvector for $T$
  2. $E_{\lambda}=\text{ker}(T-\lambda I)\neq 0$
  3. $T-\lambda I$ is not invertible
  4. $\text{det}(T-\lambda I)=0$

Concepts

Since $T-\lambda I$ connutes with $T$ and

If

Coming soon

Only If

Coming soon

Proof

Coming soon
FullPage
result
concepts
If
Only If
proof