Linear Algebra: Span, Null Space, Column Space & Linear Transformations

Introduction & Course Logistics

• Lecture opens with administrative notice on copyright (Section 113P, Copyright Act 1968); redistribution prohibited.
• Appreciation for high attendance; counters common trend of declining lecture participation.
• Current focus: Chapter 4.2 (Lei & others) – builds on MATH1013 material in a more abstract framework.
• Announcements
  • New MATLAB quiz released; multiple-choice style allows trial-and-error learning.
  • Optional textbook exercises (calculus posted by Alex, linear-algebra problems forthcoming). Some include solutions.
  • Seeking assistance encouraged whenever wording in different resources feels unfamiliar.

Recap: Vector Spaces & Subspaces

• Vector space examples: $\mathbb{R}^n$, polynomials of bounded degree, matrices of fixed size, functions, etc.
• Defining operations: vector addition + scalar multiplication must satisfy the 10 vector-space axioms.
• Subspace: non-empty subset that is closed under the above two operations; inherits the same axioms.

Linear Combinations & Span

• Scalar field fixed to $\mathbb{R}$ for this course.
• Linear combination of $v<em>1,\dots,v</em>n\in V$: $c<em>1v</em>1+\dots+c<em>n v</em>n$ with finitely many coefficients $c_i\in\mathbb{R}$.
  • Finite restriction avoids convergence issues of infinite sums.
  • In $\mathbb{R}^n$ the combination equals matrix–vector product $[v<em>1\;\dots\; v</em>n]\,\mathbf{c}$.
• Span
  • $\operatorname{span}{v<em>1,\dots,v</em>n}={c<em>1v</em>1+\dots+c<em>n v</em>n\mid c_i\in\mathbb{R}}$.
  • Convention: $\operatorname{span}(\varnothing)={\mathbf{0}}$ (analogue of $0! = 1$).
  • The span of finitely many vectors is itself a subspace and is, in fact, the smallest subspace containing those vectors.
• Example (2×2 matrices)
  • Given three matrices, their span consists of all matrices of the form $\begin{bmatrix}c<em>1+c</em>2+c<em>3 & c</em>1 \ c<em>2 & c</em>3\end{bmatrix}$, hence a subspace of $M_{2\times2}(\mathbb{R})$.

Null Space of a Matrix

• Homogeneous linear system $A\mathbf{x}=\mathbf{0}$; right-hand side is entirely zeros.
  • Always consistent because $\mathbf{x}=\mathbf{0}$ solves it.
• Definition: $\mathcal{N}(A)={\mathbf{x}\in\mathbb{R}^n\mid A\mathbf{x}=\mathbf{0}}$.
• Dimensions: for $A$ of size $m\times n$, null space lives in $\mathbb{R}^n$ (domain dimension).
• Proof that $\mathcal{N}(A)$ is a subspace (classic three-step method)
  1. Contains $\mathbf{0}$ since $A\mathbf{0}=\mathbf{0}$.
  2. Closed under addition: if $\mathbf{v},\mathbf{w}\in\mathcal{N}(A)$ then $A(\mathbf{v}+\mathbf{w})=A\mathbf{v}+A\mathbf{w}=\mathbf{0}+\mathbf{0}=\mathbf{0}$.
  3. Closed under scalar multiplication: $A(c\mathbf{v})=c\,A\mathbf{v}=c\,\mathbf{0}=\mathbf{0}$.
• Alternative proof: solution set of a homogeneous system → expressible as span of finitely many vectors (after row-reduction) → subspace.
• Worked example
  • Row-reduce augmented matrix, identify free variables $x<em>2,x</em>4,x<em>5$, express $x</em>1,x_3$ in terms of them.
  • General solution $= p\,\mathbf{u}+q\,\mathbf{v}+r\,\mathbf{w}$; hence $\mathcal{N}(A)=\operatorname{span}{\mathbf{u},\mathbf{v},\mathbf{w}}$.

Column Space of a Matrix

• Definition: $\operatorname{col}(A)=\operatorname{span}{\text{columns of }A}$ — the set of all linear combinations of columns.
• Subspace of $\mathbb{R}^m$ because each column has $m$ entries (codomain dimension).
• Set notation: $\operatorname{col}(A)={\mathbf{b}\in\mathbb{R}^m\mid \exists\,\mathbf{x}\in\mathbb{R}^n\text{ such that }A\mathbf{x}=\mathbf{b}}$.
• Example
  • $A=[\,\mathbf{v}<em>1\;\mathbf{v}</em>2\,]$, \mathbf{b}=\begin{bmatrix}1\2\3\end{bmatrix}.
  • Solve $A\mathbf{x}=\mathbf{b}$ via augmented matrix. If consistent ⇒ $\mathbf{b}\in\operatorname{col}(A)$.
  • If columns are multiples, $\operatorname{col}(A)=\operatorname{span}{\mathbf{v}<em>1}=\operatorname{span}{\mathbf{v}</em>2}$.
• Reflective questions posed to students:
  • When does $\mathcal{N}(A)={\mathbf{0}}$?
  • When does $\operatorname{col}(A)=\mathbb{R}^m$ or reduce to ${\mathbf{0}}$?

Linear Transformations: Abstract Perspective

• Notation: $T:V \to W$.
• Requirements for linearity
  1. $T(\mathbf{v}+\mathbf{w}) = T(\mathbf{v}) + T(\mathbf{w})$.
  2. $T(c\,\mathbf{v}) = c\,T(\mathbf{v})$ for all $c\in\mathbb{R}$.
• Interpretative slogan: “Applying operations in $V$ then mapping equals mapping then applying operations in $W$.”
• Remember: The "$+$" or "$c\,\cdot$" on the two sides may denote different implementations (e.g.
  polynomial addition vs. matrix addition).

Canonical Examples

• Zero map $0:V\to W$, $0(\mathbf{v})=\mathbf{0}_W$.
• Identity map $I:V\to V$, $I(\mathbf{v})=\mathbf{v}$.
• Matrix multiplication: $T(\mathbf{x})=A\mathbf{x}$ for $A\in\mathbb{R}^{m\times n}$.
  • Fundamental theorem: every linear map $\mathbb{R}^n\to\mathbb{R}^m$ can be represented by a unique matrix $A$.
• Derivative operator $D:P<em>n\to P</em>n$, $D(a<em>0+a</em>1x+\dots+a<em>n x^n)=a</em>1+2a<em>2x+\dots+n a</em>n x^{n-1}$.
• Transpose map $T:A\mapsto A^T$ on matrices $M<em>{2\times3}(\mathbb{R})\to M</em>{3\times2}(\mathbb{R})$.

Non-Examples & Pitfalls

• Mapping with products or powers of entries (e.g. T\big(\begin{bmatrix}a & b\c & d\end{bmatrix}\big)=\begin{bmatrix}ad & bc\ac & bd\end{bmatrix}) breaks linearity.
• Adding constant terms: $T(p)=x\,p(x)+1$ maps $P<em>2\to P</em>3$ but $T(0)\neq0$ ⇒ not linear.

Quick Test: $T(\mathbf{0})$

• Proven property: For any linear transformation, $T(\mathbf{0}<em>V)=\mathbf{0}</em>W$.
  • Proof sketch: $T(\mathbf{0}) = T(\mathbf{0}+\mathbf{0}) = T(\mathbf{0})+T(\mathbf{0})$ ⇒ subtract ⇒ $T(\mathbf{0})=\mathbf{0}$.
  • Contrapositive logic: If a map sends $\mathbf{0}$ to a non-zero vector, it cannot be linear.

Kernel & Range – Abstract Analogues

• Kernel (a.k.a. null space of a transformation)
  • $\ker(T)={\mathbf{v}\in V\mid T(\mathbf{v})=\mathbf{0}_W}$ — subspace of $V$.
• Range (a.k.a. image)
  • $\operatorname{ran}(T)={\mathbf{w}\in W\mid \exists\,\mathbf{v}\in V: T(\mathbf{v})=\mathbf{w}}$ — subspace of $W$.
• Connection to matrices
  • For $T(\mathbf{x})=A\mathbf{x}$: $\ker(T)=\mathcal{N}(A)$ and $\operatorname{ran}(T)=\operatorname{col}(A)$.

Proof & Reasoning Skills Emphasized

• Subspace verification via three axioms vs. span argument.
• Row-reduction to convert implicit descriptions into explicit (parametric) ones.
• Logical equivalence $A\Rightarrow B \iff \lnot B \Rightarrow \lnot A$ used for disproving linearity.
• Importance of tracking ambient spaces (domain vs. codomain) for null vs. column spaces, or kernel vs. range.

Practical & Ethical Implications

• Ethical: respect licensing; do not redistribute lecture content without permission.
• Practical: Understanding abstract definitions allows extension beyond $\mathbb{R}^n$ – crucial in functional analysis, differential equations, computer graphics, etc.
• MATLAB quizzes & optional exercises foster incremental learning; leverage trial-and-error in multiple-choice context.

Suggested Exercises & Reflections

• Determine conditions on $A\in\mathbb{R}^{m\times n}$ for:
  • $\mathcal{N}(A)={\mathbf{0}}$ (hint: pivot in every column).
  • $\operatorname{col}(A)=\mathbb{R}^m$ (hint: pivot in every row).
• Show that the transpose map $T:M<em>{m\times n}\to M</em>{n\times m}$ is linear using the two axioms explicitly.
• Provide an explicit spanning set for the kernel of $D:P<em>3\to P</em>3$.
• Explore real-world link: interpret kernel of a differential operator as the set of solutions to a homogeneous ODE.

Study Tips

• Re-derive proofs on your own; writing each line cements logic chains.
• When given an “implicit” description (e.g.
  equations), row-reduce to obtain the “explicit” span form.
• For any proposed map, test linearity rapidly by checking $T(\mathbf{0})$ and by trying to violate one axiom with simple counter-examples.
• Keep track of dimensions: domain size $n$ vs. codomain size $m$ influences which space ((\mathbb{R}^n) or (\mathbb{R}^m)) a subspace lives in.