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Linear Transformations

A linear transformation is a special function, a function that maps an object in one vector space to an object in another vector space (Note: the two vector spaces can be the same vector space.)

Let $V$ and $W$ be vector spaces with ordered bases $\beta$ and $\gamma$ respectively:

                 $\beta =\left\{\begin{array}{ccc}{x}_1& \cdots & x_{\mathrm{N}}\end{array}\right\}\gamma =\left\{\begin{array}{ccc}{y}_1& \cdots & y_{\mathrm{M}}\end{array}\right\}$

Let $T$ be a linear transformation from $V$ into $W$:

                 $T:V\to W$

The transformation $T$ takes any vector $x$ in $V$ and maps it to a vector $T\left(x\right)$ in $W$:

                 $x\to T\left(x\right)$

For example, the vectors in the basis $\beta$ are mapped as follows:

                 $x_j\to T\left(x_j\right)$

                 $T\left(x_j\right)=\sum _{i=1}^{\mathrm{M}}{a}_{ij}{y}_i 1\le j\le \mathrm{N}$

The above expression maps the basis vector $x_j$ of $V$ to the vector $T\left(x_j\right)$in $W$. The vector $T\left(x_j\right)$ is computed as a linear combination of the $\mathrm{M}$ basis vectors in the basis $\gamma$ of $W$; each basis vector contributes to the combination by an amount determined by the (unique) real number $a_{ij}$. Therefore, there are $\mathrm{M}$ such numbers for each $T\left(x_j\right)$ that corresponds to $x_j$.

The coefficients $a_{ij}$ in the linear combinations above form a matrix $A$ which we call the transformation matrix that represents $T$. Each column of this matrix is a coordinate vector that corresponds to a basis vector in the basis $\beta$ of the vector space $V$. More specifically, the $j$th column of this matrix represents the coordinate vector ${\left[T\left(x_j\right)\right]}_{\gamma }$ of $T\left(x_j\right)$ corresponding to the basis vector $x_j$:

                 $A=\left(\begin{array}{ccc}{\left[T\left(x_1\right)\right]}_{\gamma }& \cdots & {\left[T\left(x_{\mathrm{N}}\right)\right]}_{\gamma }\end{array}\right)$

We also use the notation ${\left[T\right]}_{\beta }^{\gamma }$ to denote this matrix in order to stress the fact that $T$ is a linear transformation with respect to the ordered bases $\beta$ and $\gamma$.

                 $A={\left[T\right]}_{\beta }^{\gamma }$ and $A_{ij}=a_{ij}$ where $1\le i\le \mathrm{M}$ and $1\le j\le \mathrm{N}$

The following are important facts worth remembering about the transformation matrix $A$.

Each column of $A$ represents a coordinate vector corresponding to a basis vector in $\beta$ of $V$. Namely, the $j$th column of $A$ represents the coordinate vector ${\left[T\left(x_j\right)\right]}_{\gamma }$ of $T\left(x_j\right)$ corresponding to the basis vector $x_j$ in the basis $\beta$ of $V$.

The transformation matrix $A$ has $\mathrm{N}$ columns, because its $j$th column represents the coordinate vector of $T\left(x_j\right)$ corresponding to the basis vector $x_j$ and there are $\mathrm{N}$ such basis vectors.

The transformation matrix $A$ has $\mathrm{M}$ rows, because its $j$th column represents the coordinate vector ${\left[T\left(x_j\right)\right]}_{\gamma }$ of $T\left(x_j\right)$ corresponding to the basis vector $x_j$ and the coordinate vector has $\mathrm{M}$ elements.

Examples

Click here to see examples of linear transformations.

[This content was created with SL, available on the App Store.]