The Matrix of A Linear Transformation - Linear Algebra Study Guide

The Matrix of A Linear Transformation

• Theorem 10: Existence and Uniqueness of the Standard Matrix

  • Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation.
  • There exists a unique matrix $A$ such that $T(\mathbf{x}) = A\mathbf{x}$ for all $\mathbf{x}$ in $\mathbb{R}^n$.
  • The matrix $A$ is an $m \times n$ matrix whose $j$th column is the vector $T(\mathbf{e}_j)$, where $\mathbf{e}_j$ is the $j$th column of the identity matrix in $\mathbb{R}^n$.
  • The formula for $A$ is given by:         $A = \begin{pmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \dots & T(\mathbf{e}_n) \end{pmatrix}$

• Proof of Theorem 10

  • Write $\mathbf{x} = I_n\mathbf{x} = \left[ \mathbf{e}_1 \quad \mathbf{e}_2 \quad \dots \quad \mathbf{e}_n \right] \mathbf{x} = x_1\mathbf{e}_1 + x_2\mathbf{e}_2 + \dots + x_n\mathbf{e}_n$.
  • Use the linearity of $T$ to compute:         $T(\mathbf{x}) = T(x_1\mathbf{e}_1 + \dots + x_n\mathbf{e}_n) = x_1T(\mathbf{e}_1) + \dots + x_nT(\mathbf{e}_n)$
  • By the definition of matrix-vector multiplication:         $T(\mathbf{x}) = \begin{pmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \dots & T(\mathbf{e}_n) \end{pmatrix} \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} = A\mathbf{x}$

• The Standard Matrix

  • The matrix $A$ derived in Theorem 10 is called the standard matrix for the linear transformation T.
  • Every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be viewed as a matrix transformation, and vice versa.
  • The term "linear transformation" focuses on the algebraic property of the mapping, whereas "matrix transformation" describes how the mapping is computationally implemented.

Example 2: Dilation Transformation

• Problem: Find the standard matrix $A$ for the dilation transformation $T(\mathbf{x}) = 3\mathbf{x}$ for $\mathbf{x}$ in $\mathbb{R}^2$.
• Solution:
  • Identify the standard basis vectors for $\mathbb{R}^2$:         $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$
  • Apply the transformation to these vectors:         $T(\mathbf{e}_1) = 3\mathbf{e}_1 = 3\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}$$T(\mathbf{e}_2) = 3\mathbf{e}_2 = 3\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \end{pmatrix}$
  • Form the standard matrix $A$:         $A = \begin{pmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$

Geometric Linear Transformations of R Squared

• Table 1: Reflections

  • Reflection through the $x_1$-axis:
    • Image of Unit Square: The image (depicted as a goldfish) is inverted across the horizontal axis. The point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ stays, while $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ maps to $\begin{pmatrix} 0 \\ -1 \end{pmatrix}$.
    • Standard Matrix: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
  • Reflection through the $x_2$-axis:
    • Image of Unit Square: The goldfish is mirror-imaged across the vertical axis. The point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ maps to $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$, while $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ stays.
    • Standard Matrix: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$
  • Reflection through the line $x_2 = x_1$:
    • Image of Unit Square: The goldfish is diagonally inverted. The point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ maps to $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$, and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ maps to $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$.
    • Standard Matrix: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
  • Reflection through the line $x_2 = -x_1$:
    • Image of Unit Square: The goldfish is diagonally inverted across the line $x_2 = -x_1$. The point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ maps to $\begin{pmatrix} 0 \\ -1 \end{pmatrix}$, and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ maps to $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$.
    • Standard Matrix: $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$
  • Reflection through the origin:
    • Image of Unit Square: The goldfish is diagonally inverted with diagonal sides mirrored. The point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ maps to $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$, and the point $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ maps to $\begin{pmatrix} 0 \\ -1 \end{pmatrix}$.
    • Standard Matrix: $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$

• Table 2: Contractions and Expansions

  • Horizontal Contraction and Expansion:
    • Transformation: The $x_1$-coordinate is scaled by a factor $k$.
    • Conditions: Shrunk vertically if $0 < k < 1$; expanded vertically if $k > 1$.
    • Standard Matrix: $\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}$
  • Vertical Contraction and Expansion:
    • Transformation: The $x_2$-coordinate is scaled by a factor $k$.
    • Standard Matrix: $\begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}$

• Table 3: Shears

  • Horizontal Shear:
    • Transformation: The image (bird) is skewed leftward if $k < 0$ and rightward if $k > 0$.
    • Standard Matrix: $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$
  • Vertical Shear:
    • Transformation: The image (bird) is skewed downward if $k < 0$ and upward if $k > 0$.
    • Standard Matrix: $\begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix}$

• Table 4: Projections

  • Projection onto the $x_1$-axis:
    • Transformation: The image is flattened horizontally onto the $x_1$-axis.
    • Standard Matrix: $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
  • Projection onto the $x_2$-axis:
    • Transformation: The image is flattened vertically onto the $x_2$-axis.
    • Standard Matrix: $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$

Existence and Uniqueness Questions

• Definition of Onto Mapping

  • A mapping $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is said to be onto $\mathbb{R}^m$ if each $\mathbf{b}$ in $\mathbb{R}^m$ is the image of at least one $\mathbf{x}$ in $\mathbb{R}^n$.
  • Equivalently, $T$ is onto when the range of $T$ is equal to all of the codomain $\mathbb{R}^m$.
  • For each $\mathbf{b}$, there exists at least one solution to $T(\mathbf{x}) = \mathbf{b}$. This is an existence question.
  • $T$ is not onto if there is some $\mathbf{b}$ for which the equation has no solution.

• Definition of One-to-One Mapping

  • A mapping $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is said to be one-to-one if each $\mathbf{b}$ in $\mathbb{R}^m$ is the image of at most one $\mathbf{x}$ in $\mathbb{R}^n$.
  • This is a uniqueness question.

• Theorem 11: One-to-One and Trivial Solutions

  • Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation.
  • $T$ is one-to-one if and only if the equation $T(\mathbf{x}) = \mathbf{0}$ has only the trivial solution.
  • Proof Sketch:
    • If $T$ is one-to-one, it can have at most one solution for $T(\mathbf{x}) = \mathbf{0}$. Since $T(\mathbf{0}) = \mathbf{0}$ always holds for linear transformations, that is the only solution.
    • If $T$ is not one-to-one, there exists some $\mathbf{b}$ such that $T(\mathbf{u}) = \mathbf{b}$ and $T(\mathbf{v}) = \mathbf{b}$ for $\mathbf{u} \neq \mathbf{v}$. Then $T(\mathbf{u} - \mathbf{v}) = T(\mathbf{u}) - T(\mathbf{v}) = \mathbf{b} - \mathbf{b} = \mathbf{0}$. Since $\mathbf{u} - \mathbf{v} \neq \mathbf{0}$, the equation $T(\mathbf{x}) = \mathbf{0}$ has a non-trivial solution.

• Theorem 12: Range and Uniqueness in Matrix Terms

  • Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation with standard matrix $A$.
  • a) $T$ maps $\mathbb{R}^n$ onto $\mathbb{R}^m$ if and only if the columns of $A$ span $\mathbb{R}^m$.
  • b) $T$ is one-to-one if and only if the columns of $A$ are linearly independent.

• Example 4: Analyzing Mapping Properties

  • Problem: Let $T$ be the linear transformation with standard matrix $A = \begin{pmatrix} 1 & -4 & 8 & 1 \\ 0 & 2 & -1 & 3 \\ 0 & 0 & 0 & 5 \end{pmatrix}$. Does $T$ map $\mathbb{R}^4$ onto $\mathbb{R}^3$? Is $T$ one-to-one?
  • Analysis of Onto:
    • The matrix $A$ is in echelon form and has a pivot position in every row (rows 1, 2, and 3).
    • By Theorem 4 (Section 1.4), for each $\mathbf{b}$ in $\mathbb{R}^3$, the equation $A\mathbf{x} = \mathbf{b}$ is consistent.
    • Therefore, $T$ maps $\mathbb{R}^4$ onto $\mathbb{R}^3$.
  • Analysis of One-to-One:
    • The equation $A\mathbf{x} = \mathbf{b}$ has 4 variables but only 3 pivot (basic) variables.
    • There is at least one free variable (the third variable corresponding to the column without a pivot).
    • Because there is a free variable, each $\mathbf{b}$ in the range is the image of more than one $\mathbf{x}$.
    • Therefore, $T$ is not one-to-one.