Lesson 112 — Linear transformations

$T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x, y) = (2x,\, 3y)$. Verify that $T$ is linear and find its matrix.

$T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x, y) = (x + 1,\, y)$. Why is $T$ not a linear transformation?

$T: \mathbb{R} \to \mathbb{R}$, $T(x) = x^2$. Give a concrete counterexample to show that $T$ is not linear.

$T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x, y) = (y,\, x)$. Show that it is linear and find its $2 \times 2$ matrix.

$T: \mathbb{R}^3 \to \mathbb{R}^2$, $T(x, y, z) = (x + y,\, y + z)$. Show that it is linear and find the $2 \times 3$ matrix.

$D: \mathcal{P}_3 \to \mathcal{P}_3$, $D(p) = p'$. Find the $4 \times 4$ matrix in basis $\{1, x, x^2, x^3\}$. What is special about this matrix?

$I: \mathcal{P}_2 \to \mathcal{P}_3$, $I(p)(x) = \int_0^x p(t)\,dt$. Show that it is linear and find the $4 \times 3$ matrix in canonical bases.

$T: \mathcal{M}_2 \to \mathcal{M}_2$, $T(A) = A^\top$. Show that it is linear and write the $4 \times 4$ matrix in the canonical basis of $\mathcal{M}_2$.

Fix $B \in \mathcal{M}_n$. Define $T: \mathcal{M}_n \to \mathcal{M}_n$ by $T(A) = AB$. Show that $T$ is linear.

$T: \mathcal{M}_2 \to \mathbb{R}$, $T(A) = \det A$. Is $T$ a linear transformation?

$T: \mathcal{P}_2 \to \mathcal{P}_2$, $T(p)(x) = p(2x)$. Show that it is linear and find the diagonal matrix in basis $\{1, x, x^2\}$.

$T: V \to W$, $T(v) = 0$ for all $v \in V$. Show that $T$ is linear. What is its matrix in any basis?

Find the $2 \times 2$ matrix of rotation by 45° in the plane. Verify that its determinant is 1.

Find the $2 \times 2$ matrix of reflection across line $y = x$. Verify that $[T]^2 = I$.

Find the $2 \times 2$ matrix of orthogonal projection onto the $y$-axis. Verify that $P^2 = P$ (idempotence).

Find the $2 \times 2$ matrix of orthogonal projection onto line $y = x$. Verify idempotence.

Find the matrix of non-uniform scaling $(x,y) \mapsto (3x,\, 2y)$. What is the geometric meaning of the determinant?

Find the matrix of horizontal shear with factor 2: $T(x, y) = (x + 2y,\, y)$.

Composition: rotation by 30° followed by scaling by 2. Compute the product matrix $[E_2][R_{30}]$.

Composition: reflection across line $y = x$ and then rotation by 90°. Compute the product matrix and identify the resulting transformation.

In $\mathbb{R}^3$, find the $3 \times 3$ matrix of rotation by 90° around the $z$-axis ($z$-axis stays fixed).

In $\mathbb{R}^3$, find the $3 \times 3$ matrix of orthogonal projection onto the $xy$-plane.

$T: \mathcal{P}_2 \to \mathcal{P}_2$, $T(p) = p' + p$. Find the $3 \times 3$ matrix in canonical basis.

$T: \mathbb{R}^3 \to \mathbb{R}^3$, $T(v) = v \times (1,1,1)$ (cross product with fixed $(1,1,1)$). Find the $3 \times 3$ matrix.

$T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x,y) = (x, -y)$ (reflection in $x$-axis). Find $[T]$ in canonical basis and in basis $\mathcal{B}' = \{(1,1),\,(1,-1)\}$. Confirm that they are similar.

Show that similar matrices have the same determinant and the same trace.

Show that if $A \sim B$ (similar), then $A^k \sim B^k$ for all $k \geq 1$. Conclude: nilpotence is invariant under similarity.

In computer graphics, translation by $(a, b)$ in $\mathbb{R}^2$ is not a linear transformation. How do homogeneous coordinates allow representing it as a linear transformation in $\mathbb{R}^3$? Write the $3 \times 3$ matrix.

Portfolio with $n$ assets, weights $w \in \mathbb{R}^n$, expected returns $\mu \in \mathbb{R}^n$. Is expected return $r_p = w^\top \mu$ a linear transformation in $w$? What about variance $\sigma_p^2 = w^\top \Sigma w$?

Write the Toeplitz $3 \times 5$ matrix that performs 1D linear convolution with kernel $k = (1, 2, 1)$ on a signal of length 5 (valid output, length 3).

In machine learning, a dense layer is $y = Wx + b$. Which part is a linear transformation? Why adding $b$ does not make the layer linear? Where does the non-linearity of a neural network come from?

LTI system: $\dot{x} = Ax$, solution $x(t) = e^{At}x_0$. Show that the map $x_0 \mapsto x(t)$ is a linear transformation. What is the matrix $e^{At}$?

The differentiation operator $D: \mathcal{P}_n \to \mathcal{P}_n$ has a nilpotent matrix. Explain why $D^{n+1} = 0$, and what this means in terms of polynomials.

Why is $\dim \mathcal{L}(V, W) = (\dim V)(\dim W)$? What does this say about the space of all $m \times n$ matrices?

$T(0) = 0$ is a necessary condition for linearity. Give an example of $T$ with $T(0) = 0$ that is not linear. Why is $T(0) = 0$ not sufficient?

If $A = [T]_{\mathcal{B}}$ represents $T$ in $\mathcal{B}$, explain the geometric meaning of formula $[T]_{\mathcal{B}'} = P^{-1}AP$. What does each factor do?

Explain, without computing, why matrix product $[T][S]$ is exactly composition $T \circ S$. What is the relationship between matrix product definition and composition definition?

Show that $\mathcal{L}(V, W)$ (the set of all linear transformations from $V$ to $W$) is itself a vector space, with operations $(T_1 + T_2)(v) = T_1(v) + T_2(v)$ and $(aT)(v) = a\,T(v)$.

Find $T: \mathbb{R}^2 \to \mathbb{R}^2$ with $T^2 = -I$ (negative identity). Try rotation by 90°. What is the connection with complex numbers?

Prove: every linear transformation $T: \mathbb{R} \to \mathbb{R}$ has form $T(x) = ax$ for some $a \in \mathbb{R}$.

Proof. Prove that composition of linear transformations is linear. Let $S: U \to V$ and $T: V \to W$ both be linear. Show that $T \circ S: U \to W$ is linear.

Proof. Prove by induction that every linear transformation preserves arbitrary linear combinations: $T\!\left(\sum c_i v_i\right) = \sum c_i T(v_i)$.

Proof. Prove: $T$ linear is injective $\iff \ker T = \{0\}$.

Proof. Prove the linear extension theorem: given vector spaces $V$ (dim $n$) and $W$, and vectors $w_1, \ldots, w_n \in W$ arbitrary, there exists unique linear transformation $T: V \to W$ with $T(v_i) = w_i$ for $i = 1, \ldots, n$.