Unit 04 · Algebra I

Linear Transformations

The maps that respect vector-space structure. They are the real subject of linear algebra — matrices are just the way we write them down once a basis is chosen. This unit builds the vocabulary: linearity, null space, range, and the theorem that a linear map is pinned down by its values on a basis.

By the end you can

Verify or refute linearity from the definition.
Prove the null space and range are subspaces.
Prove: injective $\Longleftrightarrow$ null space is $\{0\}$.
Use "a linear map is determined by its values on a basis."

Why it matters & where it comes from

Why it exists

Linearity — $T(av+bw)=aT(v)+bT(w)$ — is the structure-preserving condition. Studying the maps, not just the spaces, is what gives linear algebra its power.

In the real world

Every layer of a neural network is a linear transformation followed by a nonlinearity; rotations and projections in graphics are linear maps; so are the transforms of signal processing.

Story & open questions

The shift from "matrix as array of numbers" to "matrix as a linear map in coordinates" is the conceptual leap that organizes the whole subject and generalizes to infinite dimensions in functional analysis.

Definition and first properties

Definition Linear transformation

$T:V\to W$ (both spaces over the same field $F$) is linear if $$T(u+v)=T(u)+T(v)\qquad\text{and}\qquad T(av)=a\,T(v)$$ for all $u,v\in V$, $a\in F$ — equivalently, $T(au+bv)=aT(u)+bT(v)$.

Theorem

If $T$ is linear then $T(0)=0$, and $T$ preserves all linear combinations: $T(a_1v_1+\cdots+a_kv_k)=a_1T(v_1)+\cdots+a_kT(v_k)$.

Proof of $T(0)=0$

$T(0)=T(0\cdot0)=0\cdot T(0)=0$, using homogeneity with the scalar $0$ and Unit 1's identity $0w=0$. ∎

Examples and non-examples

Linear: multiplication by a fixed matrix; rotation of the plane about the origin; differentiation $D(p)=p'$ on polynomials; the trace $\operatorname{tr}:M_{n\times n}(F)\to F$; the evaluation map $p\mapsto p(2)$.

Not linear: $T(x)=x+1$ (misses $T(0)=0$); $T(x,y)=(x^2,y)$ (fails homogeneity: $T(2,0)=(4,0)\ne2\,T(1,0)$); the determinant on $M_{2\times2}$ (see Practice 8).

Null space and range

Definition

The null space (kernel) is $N(T)=\{v\in V:T(v)=0\}$; the range (image) is $R(T)=\{T(v):v\in V\}$.

Theorem

$N(T)$ is a subspace of $V$ and $R(T)$ is a subspace of $W$.

Proof

$T(0)=0$ puts $0$ in both. If $T(u)=T(v)=0$, then $T(u+av)=T(u)+aT(v)=0$: $N(T)$ passes the subspace test.

If $w_1=T(v_1)$ and $w_2=T(v_2)$, then $w_1+aw_2=T(v_1+av_2)\in R(T)$. ∎

Theorem Injectivity test

A linear $T$ is one-to-one iff $N(T)=\{0\}$.

Proof

If $T$ is one-to-one and $T(v)=0=T(0)$, then $v=0$.

If $N(T)=\{0\}$ and $T(u)=T(v)$, linearity gives $T(u-v)=0$, so $u-v\in N(T)=\{0\}$, i.e. $u=v$. ∎ One membership check replaces a two-variable argument — use it.

A linear map is determined by a basis

Theorem Existence and uniqueness

Let $\{v_1,\dots,v_n\}$ be a basis of $V$ and let $w_1,\dots,w_n$ be any vectors in $W$. Then there is exactly one linear $T:V\to W$ with $T(v_i)=w_i$ for every $i$.

Proof idea

Each $v\in V$ is uniquely $v=\sum a_iv_i$ (Unit 3); define $T(v)=\sum a_iw_i$. Uniqueness of coordinates makes $T$ well defined and linear.

Any linear map agreeing on the basis must agree on every combination — that is uniqueness. ∎

Why this theorem runs the show It says a linear map is a finite amount of data: $n$ output vectors. That is why a matrix (the coordinates of those outputs, Unit 5) can capture $T$ exactly, and why "check it on a basis" is a complete proof technique for identities between linear maps.

Common pitfalls

• Check additivity and homogeneity; each can fail without the other.
• $T(0)=0$ is necessary but not sufficient — $T(x,y)=(x^2,y)$ fixes $0$ and still fails.
• The null space lives in the domain, the range in the codomain; keep the two spaces straight.
• "Determined by a basis" needs a basis — values on a merely spanning or merely independent set do not define a map the same way.

Practice

Original problems written for this guide in the style of Friedberg, Insel & Spence exercises. Write a full solution before opening the answer.

Practice 1 · Verify

Is $T(x,y)=(x+y,\,2x)$ linear?

Answer

Yes — it is multiplication by $\left(\begin{smallmatrix}1&1\\2&0\end{smallmatrix}\right)$, and matrix multiplication is linear.

Practice 2 · Refute

Is $T(x)=x+1$ linear?

Answer

No — $T(0)=1\ne0$.

Practice 3 · Null space

Find $N(T)$ for $T(x,y)=(x+y,\,x+y)$.

Answer

$T(x,y)=0$ iff $x+y=0$: the line $\{(t,-t):t\in\mathbb{R}\}$, dimension $1$.

Practice 4 · Homogeneity fails

Show $T(x,y)=(x^2,\,y)$ is not linear.

Solution

$T(2,0)=(4,0)$ but $2\,T(1,0)=(2,0)$. One failed instance of homogeneity suffices.

Practice 5 · Injectivity test

Prove: a linear map $T$ is one-to-one iff $N(T)=\{0\}$.

Solution

($\Rightarrow$) $T(v)=0=T(0)$ forces $v=0$. ($\Leftarrow$) $T(u)=T(v)$ gives $T(u-v)=0$, so $u-v=0$.

Practice 6 · Range

Prove that $R(T)$ is a subspace of $W$.

Solution

$0=T(0)\in R(T)$; and $T(v_1)+a\,T(v_2)=T(v_1+av_2)$ shows closure under addition and scalars.

Practice 7 · Built from a basis

Find the (unique) linear $T:\mathbb{R}^2\to\mathbb{R}^2$ with $T(1,0)=(1,2)$ and $T(0,1)=(3,4)$, and give a formula for $T(x,y)$.

Solution

$T(x,y)=x(1,2)+y(3,4)=(x+3y,\,2x+4y)$ — the columns of the matrix are the images of the standard basis.

Practice 8 · Trace yes, det no

Show the trace $\operatorname{tr}:M_{2\times2}(\mathbb{R})\to\mathbb{R}$ is linear but the determinant is not.

Solution

Trace adds entrywise: $\operatorname{tr}(A+cB)=\operatorname{tr}A+c\operatorname{tr}B$. For det, $\det(2I)=4$ while $2\det I=2$ — homogeneity fails.

Expanded topic checklist

Course coverage. This unit is part of the complete MAT240H1 topic map. It should cover the following items before you move on:

Linear maps. Verify additivity and homogeneity and understand maps by their action on a basis.
Kernel and image. Use null space and range as subspaces that measure failure of injectivity and surjectivity.
Isomorphism. Recognize structure-preserving bijective linear maps.
Matrix representation preview. Translate abstract maps into coordinate matrices once bases are chosen.

Textbook/source alignment

Use the current UofT MAT240H1 calendar topics as the coverage spine, and pair lecture notes with a Friedberg-Insel-Spence-style linear algebra text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.

Core theorem map

The linear-map spine is: define on a basis, extend linearly, analyze kernel and image, then represent by a matrix after choosing bases.

Study routine. A linear transformation is completely determined by basis-vector images; exploit that before doing coordinate computations.

Deep practice extensions

Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.

Extension 1 · Linearity

Is $T(x,y)=(x+1,y)$ linear?

Guide

No. A linear map must send $(0,0)$ to $(0,0)$, but this sends it to $(1,0)$.

Extension 2 · Kernel

Find the kernel of $T(x,y,z)=(x+y,z)$.

Guide

Solve $x+y=0$ and $z=0$, so the kernel is $\{(t,-t,0):t\in F\}$.

Extension 3 · Basis images

If $T(e_1)=(1,2)$ and $T(e_2)=(3,4)$, find $T(a,b)$.

Guide

By linearity, $T(a,b)=a(1,2)+b(3,4)=(a+3b,2a+4b)$.

Extension 4 · Injectivity

State the kernel test for injectivity.

Guide

A linear map is injective iff its kernel is $\{0\}$.

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