MAT1041 · Week 10 · - Introduction to Linear Transformations

Introduction to linear transformations

You learn to test with two conditions whether a function is a linear transformation, matrix transformations, projection, reflection and rotation in the plane, abstract examples such as the derivative and trace, and how to compute a linear transformation from the images of a basis.

linear transformationmappingdomaincodomainimagepreimageadditivityhomogeneitymatrix transformationzero transformationidentity transformationprojectionreflectionrotationdifferential operator

So far we have studied vector spaces one at a time. This week we move to functions between spaces. Not every function is interesting: the functions linear algebra cares about are those that preserve addition and scalar multiplication. These are called linear transformations, and we will meet a surprising fact: every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ is in fact multiplication by a matrix. All the remaining weeks (kernel, image, transformation matrix, eigenvalue) continue this idea.

Golden rules:

First test: is $T(\mathbf{0}) = \mathbf{0}$ ? If not, it is not linear, done.
If $T(\mathbf{0}) = \mathbf{0}$ it is not enough: prove the two conditions in general or find a counterexample.
If you can write it in the form $T(\mathbf{x}) = A\mathbf{x}$ , it is linear.
If the images of a basis are given, first write the vector in that basis, then use linearity.

Definition

Let $V$ and $W$ be vector spaces. A function $T : V \to W$ is a linear transformation if for all $\mathbf{u}, \mathbf{v} \in V$ and all $c \in \mathbb{R}$

$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ (preserves addition),
$T(c\mathbf{u}) = cT(\mathbf{u})$ (preserves scalar multiplication).

$V$ is the domain, $W$ the codomain. $T(\mathbf{v})$ is the image of $\mathbf{v}$ ; all $\mathbf{v}$ with $T(\mathbf{v}) = \mathbf{w}$ are the preimage of $\mathbf{w}$ .

The two conditions in one line: $T(a\mathbf{u} + b\mathbf{v}) = aT(\mathbf{u}) + bT(\mathbf{v})$ . A linear transformation sends linear combinations to linear combinations.

Immediate properties

$T(\mathbf{0}) = \mathbf{0}$ (take $c = 0$ ).
$T(-\mathbf{v}) = -T(\mathbf{v})$ , $T(\mathbf{u} - \mathbf{v}) = T(\mathbf{u}) - T(\mathbf{v})$ .
$T(c_1\mathbf{v}_1 + \dots + c_n\mathbf{v}_n) = c_1T(\mathbf{v}_1) + \dots + c_nT(\mathbf{v}_n)$ .

The first is the most useful elimination test: if $T(\mathbf{0}) \neq \mathbf{0}$ , the transformation is not linear.

Linear or not?

Example 1 (yes). $T(x, y) = (2x - y,\ x + 3y)$ .

u = (x1,y1), v = (x2,y2)
T(u + v) = (2(x1+x2) - (y1+y2), (x1+x2) + 3(y1+y2))
         = (2x1 - y1, x1 + 3y1) + (2x2 - y2, x2 + 3y2) = T(u) + T(v)
T(cu)    = (2cx1 - cy1, cx1 + 3cy1) = c(2x1 - y1, x1 + 3y1) = cT(u)

Sample value: $T(1, 4) = (2 - 4, 1 + 12) = (-2, 13)$ .

Example 2 (no). The translation $T(x, y) = (x + 2, y)$ : $T(0, 0) = (2, 0)$ . Although shifting everything in the plane to the right looks geometrically “smooth”, it is not linear.

Transformation	Linear?	Reason
$(x, y) \mapsto (3x, x - y)$	yes	combinations with constant coefficients
$(x, y) \mapsto (x + 1, y)$	no	$T(\mathbf{0}) \neq \mathbf{0}$
$(x, y) \mapsto (x^2, y)$	no	$T(2\mathbf{e}_1) = (4, 0) \neq 2T(\mathbf{e}_1)$
$(x, y) \mapsto (xy, x)$	no	product; Q4
$(x, y) \mapsto (\lvert x \rvert, y)$	no	$T(-\mathbf{e}_1) = (1, 0) \neq -T(\mathbf{e}_1)$
$p \mapsto p'$ ( $P_n \to P_{n-1}$ )	yes	rules of differentiation
$A \mapsto A^T$	yes	$(A + B)^T = A^T + B^T$
$A \mapsto \operatorname{tr}A$	yes	the trace respects addition and scalars
$A \mapsto \det A$	no	$\det(2I) = 4 \neq 2$

PITFALL · “T(0) = 0, so it is linear”. This is only a necessary condition. $(x, y) \mapsto (xy, x)$ sends zero to zero but is not linear. Look for the counterexample in one of the two conditions.

Matrix transformations

If $A$ is an $m \times n$ matrix, then

$T(\mathbf{x}) = A\mathbf{x}$

is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ , because $A(\mathbf{u} + \mathbf{v}) = A\mathbf{u} + A\mathbf{v}$ and $A(c\mathbf{u}) = cA\mathbf{u}$ . Watch the sizes: the input space is determined by the number of columns of $A$ .

Example 3. $A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 1 & 3 \end{bmatrix}$ , $T : \mathbb{R}^3 \to \mathbb{R}^2$ :

$T(2, 1, -1) = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 1 & 3 \end{bmatrix}\begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 - 1 + 0 \\ 4 + 1 - 3 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}.$

Finding a preimage is solving a system: $A\mathbf{x} = \mathbf{w}$ . If there is no solution, the preimage of $\mathbf{w}$ is empty; if there are infinitely many, the preimage is a line or a plane.

Geometric transformations in the plane

Transformation	Matrix	Image of $(3, 5)$
Projection onto the $x$ -axis	$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$	$(3, 0)$
Reflection about the line $y = x$	$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$	$(5, 3)$
Reflection about the $x$ -axis	$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$	$(3, -5)$
Rotation by $\theta$	$\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$
Dilation by a factor $k$	$kI$	$(3k, 5k)$

A basis is enough

The most powerful property of a linear transformation: knowing the images of a basis is knowing everything. If $\{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ is a basis of $V$ , every $\mathbf{v}$ is written uniquely as $c_1\mathbf{v}_1 + \dots + c_n\mathbf{v}_n$ and

$T(\mathbf{v}) = c_1T(\mathbf{v}_1) + \dots + c_nT(\mathbf{v}_n).$

Example 4 (standard basis). $T : \mathbb{R}^3 \to \mathbb{R}^2$ , $T(\mathbf{e}_1) = (1, 2)$ , $T(\mathbf{e}_2) = (0, -1)$ , $T(\mathbf{e}_3) = (4, 1)$ .

$T(2, -1, 3) = 2(1, 2) - (0, -1) + 3(4, 1) = (2 + 0 + 12,\ 4 + 1 + 3) = (14, 8).$

If you put these images side by side as columns you get the standard matrix of $T$ : $\begin{bmatrix} 1 & 0 & 4 \\ 2 & -1 & 1 \end{bmatrix}$ . This is the topic of week 12.

If a nonstandard basis is given (Q8, Q12), first find the coefficients with a system. In Q12 we did this for a general $(x, y)$ and obtained the formula for $T$ .

Linear transformations in abstract spaces

Derivative $D : P_3 \to P_2$ , $D(a + bx + cx^2 + dx^3) = b + 2cx + 3dx^2$ .
Integral $I : P_2 \to \mathbb{R}$ , $I(p) = \int_0^1 p(x)\,dx$ .
Transpose $T : M_{m,n} \to M_{n,m}$ , $T(A) = A^T$ .
Evaluation $T : P_2 \to \mathbb{R}$ , $T(p) = p(3)$ .
Zero transformation $T(\mathbf{v}) = \mathbf{0}$ and identity transformation $T(\mathbf{v}) = \mathbf{v}$ .

The rule from calculus “the derivative of a sum is the sum of the derivatives” is really saying that the derivative is a linear transformation.

Common mistakes

Wrong	Right
Thinking $T(\mathbf{0}) = \mathbf{0}$ is sufficient	Necessary but not sufficient
“Proving” linearity with a single numerical example	Show it with general $\mathbf{u}, \mathbf{v}, c$
Writing the transformation of a $2 \times 3$ matrix as $\mathbb{R}^2 \to \mathbb{R}^3$	$\mathbb{R}^3 \to \mathbb{R}^2$ (input = number of columns)
Thinking translations and affine functions ( $x + 1$ ) are linear	A constant term breaks linearity
Using the images of a nonstandard basis as if they were the standard basis	First write the vector in that basis
Thinking $\det$ is linear	$\det(cA) = c^n\det A$
Assuming the preimage is always a single point	It can be empty, a single point or infinite

Test yourself

Exam-style questions. Solve them on paper first, then reveal the answer.

Question 1easy

Let $T(x, y) = (x + 2y,\ 3x - y)$ . Find $T(2, -1)$ .

Show answer

(0, 7)

$x = 2$ , $y = -1$ : $(2 + 2(-1),\ 3(2) - (-1)) = (0, 7)$ .

Question 2easy

Is $T : \mathbb{R}^2 \to \mathbb{R}^2$ , $T(x, y) = (x + 1,\ y)$ a linear transformation?

Show answer

A linear transformation must send zero to zero. $T(0, 0) = (1, 0) \neq (0, 0)$ .
This is a translation; adding a constant term breaks linearity.

Question 3medium

Is $T : \mathbb{R}^3 \to \mathbb{R}^2$ , $T(x, y, z) = (x - y,\ 2z)$ a linear transformation?

Show answer

Yes

$\mathbf{u} = (x_1, y_1, z_1)$ , $\mathbf{v} = (x_2, y_2, z_2)$ : $T(\mathbf{u} + \mathbf{v}) = ((x_1 + x_2) - (y_1 + y_2),\ 2(z_1 + z_2)) = T(\mathbf{u}) + T(\mathbf{v})$ . $T(c\mathbf{u}) = (cx - cy,\ 2cz) = cT(\mathbf{u})$ .
Shortcut: $T(\mathbf{x}) = A\mathbf{x}$ , $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$ .

Question 4medium

Is $T(x, y) = (xy,\ x)$ a linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$ ?

Show answer

$T(0, 0) = (0, 0)$ , this test is passed but it is not enough. $T(2(1, 1)) = T(2, 2) = (4, 2)$ , but $2T(1, 1) = 2(1, 1) = (2, 2)$ . Scalar multiplication is not preserved.
If the variables are multiplied, it is not linear.

Question 5medium

Let $T(\mathbf{v}) = A\mathbf{v}$ with $A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \end{bmatrix}$ . Find $T(1, 2, -1)$ . What are the domain and codomain of $T$ ?

Show answer

(-1, 4)

;

T : \mathbb{R}^3 \to \mathbb{R}^2

$A$ is $2 \times 3$ : the input has 3 components, the output has 2. $T : \mathbb{R}^3 \to \mathbb{R}^2$ . $A(1, 2, -1)^T = (1 + 0 - 2,\ -1 + 6 - 1) = (-1, 4)$ .

Question 6medium

Let $T(x, y) = (x + y,\ x - y)$ . Find the preimage of $\mathbf{w} = (5, 1)$ .

Show answer

(3, 2)

Preimage: all $\mathbf{v}$ with $T(\mathbf{v}) = \mathbf{w}$ . $x + y = 5$ , $x - y = 1$ .
Add: $2x = 6$ , $x = 3$ , $y = 2$ . The preimage is a single point: $\{(3, 2)\}$ .

Question 7medium

$T : \mathbb{R}^2 \to \mathbb{R}^2$ is linear with $T(1, 0) = (2, 3)$ and $T(0, 1) = (-1, 4)$ . Find $T(3, -2)$ .

Show answer

(8, 1)

$(3, -2) = 3(1, 0) - 2(0, 1)$ . Linearity: $T(3, -2) = 3T(1, 0) - 2T(0, 1) = (6, 9) - (-2, 8) = (8, 1)$ .

Question 8hard

$T : \mathbb{R}^2 \to \mathbb{R}^3$ is linear with $T(1, 1) = (1, 0, 2)$ and $T(1, -1) = (3, 2, 0)$ . Find $T(5, 1)$ .

Show answer

(9, 4, 6)

First write $(5, 1)$ in the given basis: $a(1, 1) + b(1, -1) = (5, 1)$ , $a + b = 5$ , $a - b = 1$ , so $a = 3$ , $b = 2$ . $T(5, 1) = 3T(1, 1) + 2T(1, -1) = (3, 0, 6) + (6, 4, 0) = (9, 4, 6)$ .

Question 9medium

Let $D : P_2 \to P_1$ be differentiation, $D(p) = p'$ . Find $D(3 - x + 4x^2)$ . Is $D$ linear?

Show answer

-1 + 8x

; yes

$(3 - x + 4x^2)' = -1 + 8x$ .
The rules of differentiation are exactly the linearity conditions: $(p + q)' = p' + q'$ , $(cp)' = cp'$ .

Question 10medium

Is $T : M_{2,2} \to \mathbb{R}$ , $T(A) = \det A$ a linear transformation?

Show answer

$T(2I) = \det(2I) = 4$ , but $2T(I) = 2$ . Also $\det(A + B) \neq \det A + \det B$ .
In contrast, $T(A) = \operatorname{tr}A$ and $T(A) = A^T$ are linear.

Question 11hard

Let $T$ rotate vectors in $\mathbb{R}^2$ counterclockwise by $90^\circ$ . Find $T(3, 1)$ .

Show answer

(-1, 3)

The rotation matrix is $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ , for $\theta = 90^\circ$ : $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ . $T(x, y) = (-y, x)$ , $T(3, 1) = (-1, 3)$ . Check: $(3, 1) \cdot (-1, 3) = 0$ , perpendicular.

Question 12hard

$T : \mathbb{R}^2 \to \mathbb{R}^2$ is linear with $T(1, 2) = (3, -1)$ and $T(2, 5) = (1, 1)$ . Find a formula for $T(x, y)$ .

Show answer

T(x, y) = (13x - 5y,\ -7x + 3y)

$(x, y) = a(1, 2) + b(2, 5)$ : $a + 2b = x$ , $2a + 5b = y$ . $R_2 - 2R_1$ : $b = y - 2x$ , then $a = x - 2b = 5x - 2y$ . $T(x, y) = (5x - 2y)(3, -1) + (y - 2x)(1, 1) = (13x - 5y,\ -7x + 3y)$ .
Check: $T(1, 2) = (13 - 10, -7 + 6) = (3, -1)$ .

Verification: The transformations that are linear were tested in node by checking the equalities T(u+v) = T(u)+T(v), T(cu) = cT(u) with 200 random integer vector pairs and scalars using BigInt fractions, and the nonlinear ones by computing a concrete counterexample; the images and preimages were found with exact matrix multiplication and exact solution (verify/MAT1041/10.mjs).

Review card

T: V → W is linear ⇔ (1) T(u+v) = T(u) + T(v) (2) T(cu) = cT(u) in one line: T(au + bv) = aT(u) + bT(v) Quick ELIMINATION: T(0) ≠ 0 → NOT linear. (T(0) = 0 alone is not enough!) "No" signs: constant term (x+1), product (xy), square, |x|, sin, det. "Yes" signs: every component is a sum of constant multiples of the variables → T(x) = Ax. T(x) = Ax, A m×n → T: Rⁿ → Rᵐ (input = number of COLUMNS). Image T(v); preimage: solve the system T(v) = w (none / one / infinitely many). T is COMPLETELY determined by the images of a basis: v = c1v1 + ... + cnvn → T(v) = c1T(v1) + ... + cnT(vn) Plane: rotation by θ [cos -sin; sin cos], projection onto the x-axis [1 0; 0 0], reflection about y=x [0 1; 1 0], reflection about the x-axis [1 0; 0 -1]. Abstract: derivative, integral, transpose, trace are LINEAR; det is NOT LINEAR. T(-v) = -T(v), T(u - v) = T(u) - T(v).