# Linear Transformations

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/linear-transformations

A **linear transformation** is a function $T: V \to W$ preserving vector addition and scalar multiplication: 

$$T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v}) \quad \text{and} \quad T(\alpha\vec{u}) = \alpha\,T(\vec{u})$$

Combined, this requires $T(\alpha\vec{u} + \beta\vec{v}) = \alpha\,T(\vec{u}) + \beta\,T(\vec{v})$.

Every linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ is represented by an $m\times n$ **matrix** $A$ via $T(\vec{x}) = A\vec{x}$. **Crucial rule:** The columns of $A$ are the exact images of the standard basis vectors. 

> **Pitfall:** Guessing entries without testing basis vectors is how rotations get mistaken for shears.

## Kernel, Image, and Matrices

The **kernel** $\ker(T) = \{\vec{v} : T(\vec{v})=\vec{0}\}$ is the null space. The **image** $\text{Im}(T) = \{T(\vec{v})\}$ is the column space.

$T$ is **injective** when $\ker(T) = \{\vec{0}\}$, and **surjective** when $\text{Im}(T) = W$.

| Transformation | Matrix | Geometric Effect |
| --- | --- | --- |
| Rotation by $\theta$ | $\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$ | Rotates plane |
| Reflection ($x$-axis) | $\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$ | Flips vertically |
| Scaling by $k$ | $\begin{pmatrix}k & 0 \\ 0 & k\end{pmatrix}$ | Resizes uniformly |
| Shear along $x$ | $\begin{pmatrix}1 & k \\ 0 & 1\end{pmatrix}$ | Tilts sideways |

> **Determinant:** $|\det A|$ scales areas. $\det A < 0$ reverses orientation.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Complete the description of linear transformations.

**Answer:** The **kernel** of T is the set of vectors mapped to **zero**. If the kernel is trivial ({0}), T is **injective**. The **image** of T is the set of all output vectors.

**Why:** Kernel and image characterise the structure of a linear map.

Page: https://tryals.app/practice/physics-i/linear-transformations/complete-the-description-of-linear-transformations

### 2. Sort these geometric operations into Linear or Not Linear.

**Answer:**

- Linear: Rotation, Scaling, Reflection
- Not Linear: Translation, Inversion (x$\to1/x)$

**Why:** Rotation, scaling, and reflection preserve the origin; translation and inversion do not.

Page: https://tryals.app/practice/physics-i/linear-transformations/sort-these-geometric-operations-into-linear-or-not-linear

### 3. Match each 2×2 matrix to the transformation it performs on the plane.

**Answer:**

- $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ → Rotation by 90° counterclockwise
- $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ → Reflection across the x-axis
- $\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ → Uniform scaling ×2
- $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ → Horizontal shear

**Why:** The columns-as-landed-axes trick decodes any matrix at sight: rotations swing both axes, reflections flip one, scalings stretch them, shears slide one along the other. Every 2D game engine and graphics pipeline is built from exactly these four moves.

Page: https://tryals.app/practice/physics-i/linear-transformations/match-each-2-2-matrix-to-the-transformation-it-performs-on-the-plane

### 4. Every linear transformation is determined by what it does to the basis vectors. Why is that enough to know its effect on every vector?

A. Because any vector is a combination of the basis vectors
B. Because basis vectors are the only vectors of unit length
C. Because non-basis vectors are transformed by interpolation between basis images
D. Because a transformation acts identically on all vectors of the same magnitude

**Answer:** A. Because any vector is a combination of the basis vectors

**Why:** Linearity lets the transformation pass through the combination. This is why a matrix IS just the images of the basis vectors written as columns, nothing more is needed.

Page: https://tryals.app/practice/physics-i/linear-transformations/every-linear-transformation-is-determined-by-what-it-does-to-the

### 5. Every matrix multiplication Ax defines a linear transformation.

**Answer:** True

**Why:** Matrix multiplication preserves addition and scalar multiplication, the two requirements for linearity.

Page: https://tryals.app/practice/physics-i/linear-transformations/every-matrix-multiplication-ax-defines-a-linear-transformation

### 6. Complete the statement with the correct term.

**Answer:** A linear transformation T satisfies T(u + v) = T(u) + T(v) and T(cv) = **cT(v)**.

**Why:** These two properties define linearity: preservation of addition and scaling.

Page: https://tryals.app/practice/physics-i/linear-transformations/complete-the-statement-with-the-correct-term

### 7. Every linear transformation sends the origin to the origin. Why does that rule out translation, which seems the simplest transformation of all?

A. Because linearity requires $T(0) = 0$ and translation moves it
B. Because translation cannot be written using real numbers
C. Because translations are linear, and the origin rule has exceptions
D. Because translation changes lengths, which linear maps preserve

**Answer:** A. Because linearity requires $T(0) = 0$ and translation moves it

**Why:** Translation fails the definition at a single point, the origin. Homogeneous coordinates embed the plane in 3D so that a shear there projects down to a translation here.

Page: https://tryals.app/practice/physics-i/linear-transformations/every-linear-transformation-sends-the-origin-to-the-origin-why-does