ZoomNotes for Linear Algebra

Part 8: Linear Transformations and Their Matrices

8.1 Examples of Linear Transformations

A linear transformation $T: V \to W$ satisfies:

$$T(c\mathbf{v} + d\mathbf{w}) = cT(\mathbf{v}) + dT(\mathbf{w})$$

Key requirement: $T(0) = 0$ (follows from linearity).

Examples of Linear Transformations:

1. Rotation in $\mathbb{R}^2$ by angle $\theta$: $T(x,y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)$

2. Reflection across a line

3. Projection onto a subspace

4. Diagonalization of matrices (similarity transformation $T(A) = B^{-1}AB$)

5. Derivative of polynomials: $T(p) = dp/dt$, mapping polynomials of degree $n$ to polynomials of degree $n-1$

6. Integration: $T(f) = \int_0^x f(t) dt$, mapping polynomials of degree $n$ to degree $n+1$

Composition: If $T: V \to W$ and $S: W \to Z$ are linear transformations, then $S \circ T: V \to Z$ is also linear. The matrix of the composition is the product of the individual matrices.

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8.2 Derivative Matrix $D$ and Integral Matrix $D^+$

Derivative as a linear transformation: Consider polynomials of degree $\leq 2$ (space of dimension 3) mapping to polynomials of degree $\leq 1$ (space of dimension 2).

Basis for cubics: $\{1, x, x^2\}$ Basis for linears: $\{1, x\}$

The derivative matrix $D$ (3 columns, 2 rows):

$$D = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$$

because $\frac{d}{dx}(1) = 0$, $\frac{d}{dx}(x) = 1$, $\frac{d}{dx}(x^2) = 2x$.

Integral Matrix $D^+$ (the pseudoinverse):

$$D^+ = \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1/2 \end{bmatrix}$$

because $\int_0^x 1 \, dt = x$, $\int_0^x t \, dt = x^2/2$.

Fundamental Theorem of Calculus: $D D^+ = I$ (derivative of integral = identity).

Composition: $D^+ D$ = projection onto the subspace of polynomials with zero constant term (integral of derivative loses the constant).

This matrix viewpoint shows that integration and differentiation are inverses up to a projection, exactly as in calculus.

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8.3 Basis for $V$ and Basis for $W$: Matrix for $T: V \to W$

The matrix representation $A$ of a linear transformation $T: V \to W$ depends on the choice of bases:

• Choose a basis $\mathbf{v}_1, \dots, \mathbf{v}_n$ for $V$
• Choose a basis $\mathbf{w}_1, \dots, \mathbf{w}_m$ for $W$

Column $j$ of $A$ gives the coefficients when $T(\mathbf{v}_j)$ is expressed in the output basis:

$$T(\mathbf{v}_j) = \sum_{i=1}^m a_{ij} \mathbf{w}_i$$

Applying the transformation: For any $x$ expressed in the $V$-basis, $Ax$ gives the coordinates of $T(x)$ in the $W$-basis.

Change of Basis: If we change bases, the matrix changes by a similarity transformation:

$$M = Y^{-1} A V$$

where $V$ encodes the new basis vectors for $V$ in terms of the old, and $Y$ encodes the new basis for $W$. This is a change of basis -- the same linear transformation, just described in different coordinates.

Important: The matrix size depends on the dimensions of $V$ and $W$. The rank of the matrix equals the dimension of the image of $T$ (the rank-nullity theorem).