Linear Transformations and Matrices

Linear Transformations and Matrices#

Linear Transformations#

A linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a map
\(T: \mathbb{R}^n \to \mathbb{R}^m\) that satisfies linearity:

\[ T(x + \alpha y) = T(x) + \alpha T(y) \]

for all \(x, y \in \mathbb{R}^n\) and scalars \(\alpha \in \mathbb{R}\).

A linear transformation is completely determined by its action on a basis of \(\mathbb{R}^n\).
If \(\{v_1, \dots, v_n\}\) is a basis of \(\mathbb{R}^n\), and any vector \(x\) can be written as

\[ x = \sum_{i=1}^n \alpha_i v_i \]

then

\[ T(x) = \sum_{i=1}^n \alpha_i T(v_i). \]

Examples#

  1. \(T: \mathbb{R}^2 \to \mathbb{R}^2\)
    \((1, 0) \mapsto (2, 0)\), \((0, 1) \mapsto (0, -1)\)
    Stretches the \(x\)-direction by a factor of \(2\) and flips the \(y\)-direction.

  1. \(T: \mathbb{R}^2 \to \mathbb{R}^2\)
    \((1, 0) \mapsto \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\),
    \((0, 1) \mapsto \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)
    Rotates the plane by \(\pi/4\).

  2. \(T: \mathbb{R}^2 \to \mathbb{R}\)
    \((1, 0) \mapsto 1\), \((0, 1) \mapsto 0\)
    Orthogonal projection onto the \(x\)-axis.

  3. \(T: \mathbb{R} \to \mathbb{R}^2\)
    \(1 \mapsto (1, 1)\)
    Maps \(\mathbb{R}\) onto the line \(y = x\).

Linear Transformations as Matrices#

A particularly useful basis for \(\mathbb{R}^n\) is the standard basis:

\[\begin{split} e_i = \begin{pmatrix} 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{pmatrix} \end{split}\]

with the \(1\) in the \(i\)-th position.

Any vector \(x \in \mathbb{R}^n\) can be written as:

\[ x = \sum_{i=1}^n x_i e_i. \]

If \(T(e_i) = a_i \in \mathbb{R}^m\), we can write \(a_i\) in the standard basis as:

\[\begin{split} a_i = \begin{pmatrix} a_{1i} \\ a_{2i} \\ \vdots \\ a_{mi} \end{pmatrix}. \end{split}\]

Placing these column vectors \(a_i\) side-by-side forms the matrix representation of \(T\):

\[\begin{split} A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}. \end{split}\]

Matrix-Vector Product#

For \(x \in \mathbb{R}^n\), we have:

\[ T(x) = \sum_{i=1}^n x_i a_i. \]

This is a linear combination of the columns of \(A\), with coefficients given by the entries of \(x\).
The formula for the matrix-vector product is:

\[\begin{split} Ax = \begin{pmatrix} \sum_{j=1}^n a_{1j} x_j \\ \sum_{j=1}^n a_{2j} x_j \\ \vdots \\ \sum_{j=1}^n a_{mj} x_j \end{pmatrix}. \end{split}\]

Transpose and Conjugate Transpose#

  • Transpose \(A^T\): \((i, j)\) entry of \(A^T\) is \(a_{ji}\).

  • Conjugate transpose \(A^H\): Take complex conjugate of each entry of \(A^T\).

Key property for real matrices:

\[ (Ax)^T y = x^T (A^T y). \]

For complex matrices:

\[ (Ax)^H y = x^H (A^H y). \]

Special Matrices#

  • Symmetric: \(A^T = A\)

  • Skew-symmetric: \(A^T = -A\)

  • Hermitian: \(A^H = A\)

  • Skew-Hermitian: \(A^H = -A\)

Vectors are treated as column vectors (\(n \times 1\) matrices). Their transposes are row vectors (\(1 \times n\) matrices).

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