4.1 Definition and Examples · Linear Algebra Refresher

A linear transformation is a function between vector spaces that respects the structure. The point of the chapter: every linear transformation between finite-dimensional spaces is secretly a matrix, and its kernel and range are the null space and column space of that matrix.

Definition · Linear transformation

A mapping

L : V \to W

is linear if

L (α v) = α L (v)

for every scalar

α

and

L (v_{1} + v_{2}) = L (v_{1}) + L (v_{2})

for all

v_{1}, v_{2} \in V

. When

V = W

it is called a linear operator.

The two conditions collapse into one: $L$ is linear iff $L (α v_{1} + β v_{2}) = α L (v_{1}) + β L (v_{2})$ for all scalars and vectors. By induction this extends to any linear combination, which is exactly why knowing $L$ on a basis determines $L$ everywhere.

!Exam trap

Setting

α = 0

gives

L (0_{V}) = 0_{W}

. So

L (x) = x + c

with

c \neq = 0

(a translation) is not linear, despite looking harmless. But the converse fails:

L (x) = x^{2}

sends

0 \mapsto 0

and is still not linear. Squares, roots, absolute values, and constants all break linearity.

The operators on $R^{2}$ you should recognise on sight

Operator	Formula	Standard matrix
Scaling by $α$	$L (x) = α x$	$α I$
Projection onto $x_{1}$ -axis	$L (x) = (x_{1}, 0)^{T}$	$(1000)$
Reflection in the $x_{1}$ -axis	$L (x) = (x_{1}, - x_{2})^{T}$	$(10 0 - 1)$
Rotation by $θ$ (ccw)	—	$(cos θ sin θ - sin θ cos θ)$
Swap coordinates	$L (x) = (x_{2}, x_{1})^{T}$	$(0110)$

Beyond $R^{n}$ : differentiation $D (f) = f^{'}$ on $C^{1} [a, b]$ and integration $L (f) = \int_{a}^{b} f (x) d x$ are both linear. That's the entire justification for calling systems of differential equations 'linear' in §6.2.

Kernel and range

Definition · Kernel and range

ker (L) = {v \in V : L (v) = 0_{W}}

and, for a subspace

S \subseteq V

, the image

L (S) = {L (s) : s \in S}

. The range is

L (V)

Theorem 4.1.1

L : V \to W

is linear, then

ker (L)

is a subspace of

V

and

L (S)

is a subspace of

W

for any subspace

S \subseteq V

For $L_{A} (x) = A x$ these are literally $N (A)$ and $R (A)$ — Chapter 4 is Chapter 3 with new vocabulary.

Worked example

Linearity, kernel, range

0/5 steps

Let

L : R^{3} \to R^{2}

L (x) = (x_{1} + x_{3}, x_{1} - 2 x_{2} + x_{3})^{T}

. Show

L

is linear and find

ker (L)

and the range.

★Key idea

One-to-one

⟺ ker (L) = {0}

. Onto

⟺

range

= W

. For

L_{A}

with

A

being

m \times n

: one-to-one

⟺ rank = n

; onto

⟺ rank = m

. The rank decision table from §3.6 answers both at once.

Check yourself

Which mapping is not linear?

T / F

L (0_{V}) = 0_{W}

, then

L

is a linear transformation.

L : R^{3} \to R^{2}

is defined by

L (x) = (x_{1} + x_{2}, x_{2} + x_{3})^{T}

. What is

dim ker (L)

T / F

For

L : V \to W

linear,

ker (L)

is a subspace of

V

and the range

L (V)

is a subspace of

W

4.1Definition and Examples

What you must be able to do

The operators on $R^{2}$ you should recognise on sight

Kernel and range

Linearity, kernel, range

Check yourself

4.1Definition and Examples

What you must be able to do

The operators on R2 you should recognise on sight

Kernel and range

Linearity, kernel, range

Check yourself

The operators on $R^{2}$ you should recognise on sight