$ \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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Let $V$ be a vector space over $\mathbb{F}$. A subspace of $V$ is a subset $W\subseteq V$ such that
  1. $\vec{u}+\vec{v}\in W$ for $\vec{u}, \vec{v}\in W$
  2. $\alpha\vec{u}\in W$ for $\alpha\in\mathbb{F}$, $\vec{u}\in W$
  3. 0\in V

Concepts

All subspaces are vector spaces. For any vector space $V$, $\{0\}$ and $V$ are trivial subspaces.
Notationwise, if $S_1, \dots, S_m\subseteq V$, then we write $S_1+\dots + S_m=\{x_1 + \dots + x_m:x_i\in S_i\}$
Note that a union of subspaces is also a subspace.

Used In

  • Span

Hypothesis

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Results

Coming soon
Let $V$ be a vector space over $\mathbb{F}$. A subspace of $V$ is a subset $W\subseteq V$ such that
  1. $\vec{u}+\vec{v}\in W$ for $\vec{u}, \vec{v}\in W$
  2. $\alpha\vec{u}\in W$ for $\alpha\in\mathbb{F}$, $\vec{u}\in W$
  3. 0\in V

Concepts

All subspaces are vector spaces. For any vector space $V$, $\{0\}$ and $V$ are trivial subspaces.
Notationwise, if $S_1, \dots, S_m\subseteq V$, then we write $S_1+\dots + S_m=\{x_1 + \dots + x_m:x_i\in S_i\}$
Note that a union of subspaces is also a subspace.

Used In

  • Span

Hypothesis

Coming soon

Results

Coming soon
FullPage
definition
concepts
used in
hypothesis
results
FullPage
definition
concepts
used in
hypothesis
results
Let $V$ be a vector space over $\mathbb{F}$. A subspace of $V$ is a subset $W\subseteq V$ such that
  1. $\vec{u}+\vec{v}\in W$ for $\vec{u}, \vec{v}\in W$
  2. $\alpha\vec{u}\in W$ for $\alpha\in\mathbb{F}$, $\vec{u}\in W$
  3. 0\in V

Concepts

All subspaces are vector spaces. For any vector space $V$, $\{0\}$ and $V$ are trivial subspaces.
Notationwise, if $S_1, \dots, S_m\subseteq V$, then we write $S_1+\dots + S_m=\{x_1 + \dots + x_m:x_i\in S_i\}$
Note that a union of subspaces is also a subspace.

Used In

  • Span

Hypothesis

Coming soon

Results

Coming soon
Let $V$ be a vector space over $\mathbb{F}$. A subspace of $V$ is a subset $W\subseteq V$ such that
  1. $\vec{u}+\vec{v}\in W$ for $\vec{u}, \vec{v}\in W$
  2. $\alpha\vec{u}\in W$ for $\alpha\in\mathbb{F}$, $\vec{u}\in W$
  3. 0\in V

Concepts

All subspaces are vector spaces. For any vector space $V$, $\{0\}$ and $V$ are trivial subspaces.
Notationwise, if $S_1, \dots, S_m\subseteq V$, then we write $S_1+\dots + S_m=\{x_1 + \dots + x_m:x_i\in S_i\}$
Note that a union of subspaces is also a subspace.

Used In

  • Span

Hypothesis

Coming soon

Results

Coming soon
FullPage
definition
concepts
used in
hypothesis
results