Linear Transformations, Subspaces, Independence & Bases

Linear Transformations: Kernel & Range

• We study an abstract linear transformation $T:V\to W$.
  • $\operatorname{Ker}T$ (kernel/null–space) := ${v\in V\mid T(v)=0_W}$.
  • $\operatorname{Range}T$ (image/column–space) := ${T(v)\mid v\in V}\subseteq W$.
• Both kernel and range are subspaces of their parent spaces, exactly as $\ker A$ and $\operatorname{col}A$ are for a matrix $A$.

Why $\operatorname{Range}T$ is a Subspace of $W$

• Containment: By definition every element of $\operatorname{Range}T$ lives in $W$.

• Non-emptiness/zero:

  • $T(0<em>V)=0</em>W$ (linearity).
  • So $0_W\in\operatorname{Range}T$.

• Closed under addition:

  • If $w<em>1,w</em>2\in\operatorname{Range}T$, $\exists v<em>1,v</em>2\in V$ such that $T(v<em>1)=w</em>1$ and $T(v<em>2)=w</em>2$.
  • $T(v<em>1+v</em>2)=T(v<em>1)+T(v</em>2)=w<em>1+w</em>2\in\operatorname{Range}T$.

• Closed under scalar multiplication (left as analogous exercise).

• Extra result to try: $T$ is injective $\iff$ $\ker T={0}$.

Visual Intuition: Kernel vs Range

• Draw two blobs (sets) labelled $V$ and $W$.
  • Arrows represent $T$.
• Kernel: subspace inside $V$ whose every element arrows to $0_W$.
• Range: subspace inside $W$ that receives arrows from all of $V$.

Worked Examples of Kernel & Range

• Zero transformation $T_0(v)=0$ for all $v\in V$
  • $\ker T_0=V$ (everything maps to $0$).
  • $\operatorname{Range}T<em>0={0</em>W}$.
• Differentiation $D: P<em>n\to P</em>n,\; D(p)=p'$
  • $\ker D$ = constant polynomials.
  • $\operatorname{Range}D=P_{n-1}$ (all polynomials of degree $\le n-1$; every such polynomial has an antiderivative of degree $\le n$).
• Transpose $T: M<em>{2\times3}\to M</em>{3\times2},\; T(A)=A^T$
  • $\ker T={0}$ because $A^T=0\Rightarrow A=0$.
  • $\operatorname{Range}T=M_{3\times2}$ (every $3\times2$ matrix is a transpose of a $2\times3$ one).

Linear Independence

• For vectors $v<em>1,\dots,v</em>n\in V$ we study the equation $c<em>1v</em>1+\dots+c<em>nv</em>n=0.$
  • Only–trivial–solution (all $c_i=0$) $\Rightarrow$ set is linearly independent (LI).
  • Any non–trivial solution $\Rightarrow$ set is linearly dependent (LD).
• Conventions
  • Empty set is LI (technical convenience).
  • Independence/dependence is a property of a set, not an individual vector.

Special Cases & Quick Tests

• Single–vector set ${v}$:
  • LI $\iff v\ne0$, LD otherwise.
• Any set containing $0$ is automatically LD (take coefficient of $0$ to be non-zero while others are zero).

Determining Independence in Non-$\mathbb{R}^n$ Spaces

• Example in $P_2$ (polynomials $\deg\le2$):
  • Test set ${1+2t,\;4t^2+2t+3,\;t^2+t+1}$.
  • Form equation $a(1+2t)+b(4t^2+2t+3)+c(t^2+t+1)=0.$
  • Collect coefficients (t^2,t,1):
    $\begin{cases}4b+c=0\ 2a+2b+c=0\ a+3b+c=0\end{cases}$
  • Matrix representation reduces to identity $\Rightarrow$ only trivial solution $\Rightarrow$ LI.
• Moral: translate abstract problem to homogeneous linear system in $\mathbb{R}^n$.

Bases: Concept & Properties

• Basis of vector space $V$ (finite case) = set ${v<em>1,\dots,v</em>n}$ such that
  1. LI.
  2. Spans $V$.
• Interpretations
  • “Smallest” spanning set (no redundant vectors).
  • “Largest” LI set (cannot add any new vector without losing independence).
• Properties
  • Every (non-zero) vector space has infinitely many bases; exception: ({0}) whose only basis is the empty set.
  • For spaces with finite bases, all bases contain the same number of vectors – the dimension (topic for next lecture).
  • Bases give compact computer-friendly descriptions of infinite sets.

Standard Bases Examples

• $\mathbb{R}^n$: $e<em>1=(1,0,\dots),\ e</em>2=(0,1,0,\dots),\dots,e_n$.
• $M_{2\times3}$: six matrices each with a single 1 in one entry, 0 elsewhere.

Bases in $P_2$: Candidate Sets

• Standard basis ${1,\;t,\;t^2}$ is a basis (LI and spans).
• Set ${1+t,\;t+t^2,\;t^2+1}$ is NOT a basis; fails LI or spanning (check by solving combination equation or comparing dimensions).
• Challenge: build alternative bases (e.g. ${1+t,\;t,\;t^2}$ etc.).

Bases of Null Space & Column Space of a Matrix

• Given matrix $A$, row-reduce to RREF $B$ (same solutions for $Ax=0$).

Null Space

• Solving $Ax=0$ yields general solution in terms of free variables → express as linear combination of special solution vectors.
• Those special vectors are automatically LI and span $\ker A$ ⇒ they form a basis.

Column Space

• Row operations change individual columns but preserve linear relations among them.
• Pivot columns in RREF $B$ correspond to a subset of original columns in $A$ that are LI and that generate every other column.
• Therefore: columns of $A$ whose counterparts become pivot columns in RREF form a basis of $\operatorname{col}A$.
  • Key step: AFTER locating pivot indices, go back to the original matrix to pick the basis vectors.

Constructing a Basis: Two Viewpoints

1. Trim a spanning set (Spanning Set Theorem)
   • Start with a set known to span $V$.
   • Place vectors as columns, row-reduce.
   • Remove any column that is a linear combination of preceding ones (non-pivot columns).
   • Remaining columns give a basis.
2. Extend an independent set (Basis Extension Theorem, optional)
   • Begin with LI set.
   • Keep adding vectors outside its span, preserving independence, until the set spans $V$.
   • Procedure always terminates in finite-dimensional spaces.
   • Not examinable but conceptually complementary.

Connections, Implications & Real-World Notes

• Kernel–image picture underlies fundamental theorem of linear mappings (dimension theorem to come).
• Differentiation, transposition etc. illustrate how linear algebra unifies diverse operations.
• Bases enable storage of functions, signals, or geometric data in computers (e.g. Fourier basis, wavelets).
• Ethical/practical note: handling copyrighted lecture materials must respect legal constraints (opening disclaimer).