Linear Algebra: Sets of Linear Transformations

Sets of Linear Transformations


Let $\vec{x}$ be a non-zero $n$ dimensional vector and $\vec{b}$ be a $m$ dimensional vector. We want to find the set of all possible linear transformations such that $T(\vec{x}) = A \vec{x} = \vec{b}$ where $A$ is an $m$ by $n$ matrix.


One particular solution $A_0$ to the equation $T(\vec{x}) = A \vec{x} = \vec{b}$ can be found using the outer product of $\vec{b}$ and $\vec{x}$:


$$ A_0 = \frac{\vec{b} \vec{x}^\top}{\vec{x}^\top \vec{x}} \quad \Rightarrow \quad A_0 x = \vec{b} \frac{\vec{x}^\top \vec{x}}{\vec{x}^\top \vec{x}} = \vec{b} $$

The general solution $A$ can now be expressed as:


$$ A = A_0 + N \quad \text{where} \quad N \vec{x}=\vec{0} $$

Where $N$ is an $m$ by $n$ matrix since:


$$ A\vec{x} = (A_0 + N)\vec{x} = A_0\vec{x} + N\vec{x} = \vec{b} + \vec{0} = \vec{b} $$

Let $V$ be an $n$ by $n-1$ matrix with column vectors that form an $n-1$ dimensional basis for vectors that are orthogonal to $\vec{x}$. That is,


$$ V^\top \vec{x} = \vec{0} $$

Then let $C$ be an $m$ by $n-1$ matrix containing $m(n-1)$ free variables. Then,


$$ N = C V^\top \quad \Rightarrow \quad N \vec{x} = C V^\top \vec{x} = \vec{0} $$

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