Row Equivalent Matrices Have The Same Solution - Neutrinos are the coolest particles

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Two homogeneous equations have the same solution if they are row equivalent

For two $m\times n$ matrices $A$, $B$, $Ax=0$, $Bx=0$, the two have the same solutions if $A, B$ are row equivalent.

Concepts

Coming soon

Hypothesis

We require that $Ax=0$ and $Bx=0$ as we need a solution to $Ax=0$ and assume that $B$ is obtained from $A$ by a single elementary row operation. Row operations on a column of $0$ remain the same column of $0$.
We require $A$ and $B$ to be row equivalent for $Ax=0$ and $Bx=0$ to be equivalent, in order to call on another theorem. For $A$ and $B$ to be row equivalent, there exists elementary matrices $E_1, \dots, E_k$ such that $B=E_k\cdot \dots E_1A$.

Results

If two systems have the same solution, we can solve one system by solving the other.

Proof

Suppose $B$ is obtained from $A$ using a finite number of elemenetary row operations. That is, $$\begin{align*} A=A_0\to A_1\to\dots\to A_k=B. \end{align*}$$ It suffices to show that $A_{i}x=0$ and $A_{i+1}x=0$ have solutions for $0\leq 1\leq k-1$ (yeah idk what's going on there). We can also assume that $B$ is obtained from $A$ by a single elemenetary row operation.
Since elementary row operations are all invertible using elementary row operations, then rows of $A$ are linear combinations of rows of $B$, meaning $Ax=0$ and $Bx=0$ are equivalent. By a a theorem that equivalent matrices have the same solution, $A$ and $B$ have the same solution.

Two homogeneous equations have the same solution if they are row equivalent

For two $m\times n$ matrices $A$, $B$, $Ax=0$, $Bx=0$, the two have the same solutions if $A, B$ are row equivalent.

Concepts

Coming soon

Hypothesis

We require that $Ax=0$ and $Bx=0$ as we need a solution to $Ax=0$ and assume that $B$ is obtained from $A$ by a single elementary row operation. Row operations on a column of $0$ remain the same column of $0$.
We require $A$ and $B$ to be row equivalent for $Ax=0$ and $Bx=0$ to be equivalent, in order to call on another theorem. For $A$ and $B$ to be row equivalent, there exists elementary matrices $E_1, \dots, E_k$ such that $B=E_k\cdot \dots E_1A$.

Results

If two systems have the same solution, we can solve one system by solving the other.

Proof

Suppose $B$ is obtained from $A$ using a finite number of elemenetary row operations. That is, $$\begin{align*} A=A_0\to A_1\to\dots\to A_k=B. \end{align*}$$ It suffices to show that $A_{i}x=0$ and $A_{i+1}x=0$ have solutions for $0\leq 1\leq k-1$ (yeah idk what's going on there). We can also assume that $B$ is obtained from $A$ by a single elemenetary row operation.
Since elementary row operations are all invertible using elementary row operations, then rows of $A$ are linear combinations of rows of $B$, meaning $Ax=0$ and $Bx=0$ are equivalent. By a a theorem that equivalent matrices have the same solution, $A$ and $B$ have the same solution.

FullPage

result

concepts

hypothesis

implications

proof

FullPage

result

concepts

hypothesis

implications

proof

Two homogeneous equations have the same solution if they are row equivalent

For two $m\times n$ matrices $A$, $B$, $Ax=0$, $Bx=0$, the two have the same solutions if $A, B$ are row equivalent.

Concepts

Coming soon

Hypothesis

We require that $Ax=0$ and $Bx=0$ as we need a solution to $Ax=0$ and assume that $B$ is obtained from $A$ by a single elementary row operation. Row operations on a column of $0$ remain the same column of $0$.
We require $A$ and $B$ to be row equivalent for $Ax=0$ and $Bx=0$ to be equivalent, in order to call on another theorem. For $A$ and $B$ to be row equivalent, there exists elementary matrices $E_1, \dots, E_k$ such that $B=E_k\cdot \dots E_1A$.

Results

If two systems have the same solution, we can solve one system by solving the other.

Proof

Suppose $B$ is obtained from $A$ using a finite number of elemenetary row operations. That is, $$\begin{align*} A=A_0\to A_1\to\dots\to A_k=B. \end{align*}$$ It suffices to show that $A_{i}x=0$ and $A_{i+1}x=0$ have solutions for $0\leq 1\leq k-1$ (yeah idk what's going on there). We can also assume that $B$ is obtained from $A$ by a single elemenetary row operation.
Since elementary row operations are all invertible using elementary row operations, then rows of $A$ are linear combinations of rows of $B$, meaning $Ax=0$ and $Bx=0$ are equivalent. By a a theorem that equivalent matrices have the same solution, $A$ and $B$ have the same solution.

Two homogeneous equations have the same solution if they are row equivalent

For two $m\times n$ matrices $A$, $B$, $Ax=0$, $Bx=0$, the two have the same solutions if $A, B$ are row equivalent.

Concepts

Coming soon

Hypothesis

We require that $Ax=0$ and $Bx=0$ as we need a solution to $Ax=0$ and assume that $B$ is obtained from $A$ by a single elementary row operation. Row operations on a column of $0$ remain the same column of $0$.
We require $A$ and $B$ to be row equivalent for $Ax=0$ and $Bx=0$ to be equivalent, in order to call on another theorem. For $A$ and $B$ to be row equivalent, there exists elementary matrices $E_1, \dots, E_k$ such that $B=E_k\cdot \dots E_1A$.

Results

If two systems have the same solution, we can solve one system by solving the other.

Proof

Suppose $B$ is obtained from $A$ using a finite number of elemenetary row operations. That is, $$\begin{align*} A=A_0\to A_1\to\dots\to A_k=B. \end{align*}$$ It suffices to show that $A_{i}x=0$ and $A_{i+1}x=0$ have solutions for $0\leq 1\leq k-1$ (yeah idk what's going on there). We can also assume that $B$ is obtained from $A$ by a single elemenetary row operation.
Since elementary row operations are all invertible using elementary row operations, then rows of $A$ are linear combinations of rows of $B$, meaning $Ax=0$ and $Bx=0$ are equivalent. By a a theorem that equivalent matrices have the same solution, $A$ and $B$ have the same solution.

FullPage

result

concepts

hypothesis

implications

proof