$ \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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result
concepts
hypothesis
implications
proof

Sufficient conditions for nontrivial solutions

If $A$ is a $m\times n$ matrix such that $m\lt n$, then $Ax=0$ has a nontrivial solution.

Concepts

We use row equivalence

Hypothesis

We require $m\lt n$ for the $n-m$ free variables.

Results

Coming soon

Proof

Let $R$ be the row reduced echelon form that is row equivalent to $A$. Then there are at most $m$ isolated variables. Since $n\gt m$, then there exists at least $n-m\gt 0$ free variables.

Sufficient conditions for nontrivial solutions

If $A$ is a $m\times n$ matrix such that $m\lt n$, then $Ax=0$ has a nontrivial solution.

Concepts

We use row equivalence

Hypothesis

We require $m\lt n$ for the $n-m$ free variables.

Results

Coming soon

Proof

Let $R$ be the row reduced echelon form that is row equivalent to $A$. Then there are at most $m$ isolated variables. Since $n\gt m$, then there exists at least $n-m\gt 0$ free variables.
FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

Sufficient conditions for nontrivial solutions

If $A$ is a $m\times n$ matrix such that $m\lt n$, then $Ax=0$ has a nontrivial solution.

Concepts

We use row equivalence

Hypothesis

We require $m\lt n$ for the $n-m$ free variables.

Results

Coming soon

Proof

Let $R$ be the row reduced echelon form that is row equivalent to $A$. Then there are at most $m$ isolated variables. Since $n\gt m$, then there exists at least $n-m\gt 0$ free variables.

Sufficient conditions for nontrivial solutions

If $A$ is a $m\times n$ matrix such that $m\lt n$, then $Ax=0$ has a nontrivial solution.

Concepts

We use row equivalence

Hypothesis

We require $m\lt n$ for the $n-m$ free variables.

Results

Coming soon

Proof

Let $R$ be the row reduced echelon form that is row equivalent to $A$. Then there are at most $m$ isolated variables. Since $n\gt m$, then there exists at least $n-m\gt 0$ free variables.
FullPage
result
concepts
hypothesis
implications
proof