Standard Matrix of a Linear Transformation

Definition

The standard matrix of a linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is the matrix $A \in \mathbb{R}^{m \times n}$ such that for any vector $\vec{v} \in \mathbb{R}^n$:

This matrix represents the transformation relative to the standard bases of $\mathbb{R}^n$ and $\mathbb{R}^m$.

Constructing the Standard Matrix

To find the standard matrix of a linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$:

1. Let ${\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n}$ be the standard basis of $\mathbb{R}^n$, where $\vec{e}_i$ has 1 in the $i$-th position and 0 elsewhere.

2. Compute the images of each basis vector: $T(\vec{e}_1), T(\vec{e}_2), \ldots, T(\vec{e}_n)$.

3. Form the matrix $A$ by using these images as columns:

That is, the $j$-th column of $A$ is the vector $T(\vec{e}_j)$.

For General Vector Spaces

For linear transformations between general vector spaces $T: V \rightarrow W$ with chosen bases $\mathcal{B}$ for $V$ and $\mathcal{C}$ for $W$, the process is similar:

1. Let $\mathcal{B} = {\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n}$ be a basis for $V$.

2. Compute the images: $T(\vec{v}_1), T(\vec{v}_2), \ldots, T(\vec{v}_n)$ in $W$.

3. Express each $T(\vec{v}_j)$ as a linear combination of the basis vectors in $\mathcal{C}$.

4. Use the coefficients from these linear combinations to form the columns of the matrix.

Examples

Example 1: Reflection About the y-axis

Consider the transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that reflects points about the y-axis.

The standard basis vectors of $\mathbb{R}^2$ are:

Under reflection about the y-axis:

Therefore, the standard matrix of $T$ is:

Example 2: Linear Transformation Given by a Formula

Consider $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ defined by:

The standard basis vectors of $\mathbb{R}^3$ are:

Computing the images:

Therefore, the standard matrix is:

Properties

1. Linearity: The standard matrix representation preserves all the properties of the original linear transformation.

2. Composition: If $T: U \rightarrow V$ and $S: V \rightarrow W$ are linear transformations with standard matrices $A$ and $B$ respectively, then the standard matrix of the composition $S \circ T$ is the product $BA$.

3. Inverse: If $T: V \rightarrow V$ is invertible with standard matrix $A$, then the standard matrix of $T^{-1}$ is $A^{-1}$.

4. Null Space and Image: The null space and image of the transformation $T$ correspond to the null space and column space of its standard matrix $A$.

Applications

1. Geometric Transformations: Standard matrices can represent rotations, reflections, projections, and other geometric transformations.

2. Change of Basis: When changing from one basis to another, the standard matrix helps transform the coordinates.

3. Solving Systems: The standard matrix allows us to convert problems about linear transformations into problems about systems of linear equations.

4. Computer Graphics: Standard matrices are used to implement transformations in 2D and 3D graphics.

Exercise

Find the standard matrix of the linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ defined by:

Then, find the null space and image of $T$, and verify the Rank-Nullity Theorem.