# Linear Transformations

Mathematics I · Linear Algebra · https://tryals.app/learn/mathematics-i/linear-transformations

## Structure-Preserving Maps

A **linear transformation** is a map $T : V \to W$ between vector spaces that **preserves vector addition and scalar multiplication**:
$$T(u + v) = T(u) + T(v), \quad T(cv) = c\,T(v).$$

Rotations, reflections, scalings, projections, differentiation, and integration are all linear; bending a line or adding a constant shift is not.

An immediate consequence is that **fixes the origin** ($T(\mathbf{0}) = \mathbf{0}$ by setting $c = 0$). Any map moving the origin is **not** linear.

| Property | Linear ($T(v) = Av$) | Non-Linear ($T(v) = v + b$) |
| --- | --- | --- |
| Origin | $T(\mathbf{0}) = \mathbf{0}$ | $T(\mathbf{0}) = b \neq \mathbf{0}$ |
| Addition | $T(u+v) = T(u)+T(v)$ | Fails |
| Scaling | $T(cv) = cT(v)$ | Fails |

## Matrices and Composition

**A linear transformation is completely determined by its action on a basis.** Since every vector is a combination of basis vectors, knowing $T(e_1), T(e_2), \dots$ pins down $T$ everywhere.

For finite-dimensional spaces, we form the **matrix representation** by placing the images $T(e_j)$ as the **columns** of a matrix $A$, yielding **$T(\mathbf{v}) = A\mathbf{v}$**.

Because transformations are matrices, **composing** transformations corresponds to **multiplying** matrices. Since order matters (rotate-then-reflect $\neq$ reflect-then-rotate), matrix multiplication is non-commutative.

> **Common pitfall:** Assuming **every** map is linear. **Linearity is strict:** both rules must hold simultaneously, forcing $T(\mathbf{0}) = \mathbf{0}$.

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Every linear map is fixed by what it does to a basis, and a matrix is just that record. Why does the *same* transformation have many different matrices?

A. Because a transformation can be written in different but equivalent formulas
B. Because the matrix depends on which vector it is applied to
C. Because rounding produces slightly different entries each time
D. Because the matrix records the map relative to a chosen basis

**Answer:** D. Because the matrix records the map relative to a chosen basis

**Why:** The map is geometric and the matrix is a description in a coordinate system, so changing coordinates changes the entries and not the action. That is what similarity means, and it is why diagonalisation is worth doing: you are hunting for the basis in which the description is simplest.

Page: https://tryals.app/practice/mathematics-i/linear-transformations/every-linear-map-is-fixed-by-what-it-does-to-a-basis-and-a-matrix-is

### 2. T is a linear map from three-dimensional space into the plane. Select every statement that MUST be true.

A. T(2u) = 2 T(u) for every u
B. T(u + v) = T(u) + T(v) for all u and v
C. T sends (0, 0, 0) to (0, 0)
D. T(u - v) = T(u) - T(v) for all u and v
E. T is represented by a 3x2 matrix

**Answer:** A. T(2u) = 2 T(u) for every u; B. T(u + v) = T(u) + T(v) for all u and v; C. T sends (0, 0, 0) to (0, 0); D. T(u - v) = T(u) - T(v) for all u and v

**Why:** Linearity gives the origin, scaling, additivity and hence subtraction. But the matrix is 2x3, not 3x2: it has one column per input coordinate and one row per output coordinate. Reversing the convention is the standard shape error.

Page: https://tryals.app/practice/mathematics-i/linear-transformations/t-is-a-linear-map-from-three-dimensional-space-into-the-plane-select

### 3. A linear map on the plane is completely determined once you know what it does to (1,0) and (0,1) - two vectors out of infinitely many. Why do those two pin down every other value?

A. Because the plane has two dimensions, so any two distinct vectors determine a linear map.
B. Because a linear map is continuous, so its values elsewhere follow from its values at two points by interpolation.
C. Because (1,0) and (0,1) are the only vectors of length 1 along the axes, and linear maps are defined by their effect on unit vectors.
D. Because every vector is a combination of them, and linearity respects that.

**Answer:** D. Because every vector is a combination of them, and linearity respects that.

**Why:** Write the vector in the basis and let linearity do the rest: (4,7) = 4(1,0) + 7(0,1), so two known outputs determine the third. This is what makes matrices possible, storing the images of the basis vectors as columns captures an infinite map in finitely many numbers. The two must be independent: T(1,0) and T(2,0) tell you almost nothing.

Page: https://tryals.app/practice/mathematics-i/linear-transformations/a-linear-map-on-the-plane-is-completely-determined-once-you-know-what

### 4. If T is linear and T(u) = T(v) for two different vectors u and v, then T(u - v) is the zero vector.

**Answer:** True

**Why:** True: T(u - v) = T(u) - T(v) = 0, and u - v is nonzero because u and v differ. So a linear map that collides on two inputs necessarily sends some nonzero vector to zero, this is exactly the kernel, and it is why injectivity will reduce to a kernel condition in the next lesson.

Page: https://tryals.app/practice/mathematics-i/linear-transformations/if-t-is-linear-and-t-u-t-v-for-two-different-vectors-u-and-v

### 5. Order the steps of finding the matrix of a linear transformation of the plane.

**Answer:**

1. Apply T to the first standard basis vector (1, 0)
2. Write the result as the first column of the matrix
3. Apply T to the second standard basis vector (0, 1)
4. Write that result as the second column
5. Check by computing the matrix times a test vector and comparing with T of that vector

**Why:** Because every vector is a unique combination of the basis vectors and T preserves combinations, the images of the basis vectors pin down T everywhere. Placing them as columns gives a matrix A with T(v) = Av, a matrix simply IS a linear map written in coordinates.

Page: https://tryals.app/practice/mathematics-i/linear-transformations/order-the-steps-of-finding-the-matrix-of-a-linear-transformation-of