Linear Transformations, Matrix Representation, Change of Basis & Rank–Nullity
TRB Mathematics — Interactive MCQ Quiz (50 Questions) | ARO Study Circle
📘 Study Material & Formula Cheat Sheet (click to expand)
1. Prerequisites & Intuition
Prerequisites: sets & functions, fields (ℝ, ℂ, ℚ), matrices & row reduction, systems of equations, vector spaces and basis.
Intuition: A linear transformation preserves addition and scalar multiplication. Matrix representation converts a map to a concrete array once bases are chosen. Change of basis changes only the description. Rank–Nullity is a conservation law: information in = information lost (kernel) + information preserved (image).
2. Key Definitions
Linear transformation: \(T(c\mathbf u+\mathbf v)=cT(\mathbf u)+T(\mathbf v)\) for all vectors and scalars; in particular \(T(\mathbf 0)=\mathbf 0\).
Kernel: \(\ker T=\{\mathbf v:T(\mathbf v)=\mathbf 0\}\subseteq V\). Image: \(\operatorname{Im}T=\{T(\mathbf v)\}\subseteq W\).
Rank \(=\dim(\operatorname{Im}T)\); Nullity \(=\dim(\ker T)\).
Matrix representation: For ordered bases \(B\) of \(V\) and \(G\) of \(W\), the \(j\)-th column of \([T]_B^G\) is \([T(\mathbf v_j)]_G\). Fundamental relation: \([T(\mathbf v)]_G=[T]_B^G[\mathbf v]_B\).
Change of basis: If \([\mathbf v]_B=P[\mathbf v]_{B'}\), then \([T]_{B'}=P^{-1}[T]_BP\). Two matrices are similar if \(B=P^{-1}AP\).
3. Key Theorems
- Rank–Nullity: \(\operatorname{nullity}(T)+\operatorname{rank}(T)=\dim V\) (requires \(\dim V<\infty\)).
- Matrix isomorphism: \(L(V,W)\cong M_{m\times n}(F)\); \(\dim L(V,W)=\dim V\cdot\dim W=mn\).
- Composition: \([U\circ T]=[U][T]\) for compatible bases; \(\operatorname{rank}(S\circ T)\le\min(\operatorname{rank}S,\operatorname{rank}T)\).
- Change of basis: \([T]_{B'}=P^{-1}[T]_BP\); similar matrices share det, tr, rank, char. poly, eigenvalues.
- Injectivity/surjectivity: injective \(\iff\ker T=\{\mathbf 0\}\); surjective \(\iff\operatorname{Im}T=W\). For square finite-dim: injective \(\iff\) surjective \(\iff\) invertible \(\iff\det\neq0\).
4. Exam Shortcuts
- S1. Test \(T(\mathbf 0)=\mathbf 0\) and scan for nonlinear terms (squares, products, constants).
- S2. Matrix columns = images of source-basis vectors in target-basis coordinates.
- S3. For change of basis, verify column-by-column using \(T(\mathbf v'_j)\); use trace as a check.
- S4. Surjective \(\Rightarrow\) nullity \(=\dim V-\dim W\) immediately.
- S5. Square operator: one determinant check settles injectivity, surjectivity, invertibility.
- S6. Composition: \(\ker T\subseteq\ker(S\circ T)\); \(\operatorname{Im}(S\circ T)\subseteq\operatorname{Im}S\).
- S7. Nilpotent on \(n\)-dim space: \(T^n=0\), rank \(
- S8. Idempotent \(T^2=T\): \(V=\ker T\oplus\operatorname{Im}T\); eigenvalues \(0,1\).
5. Formula Cheat Sheet
| Concept | Formula / Condition |
|---|---|
| Rank–Nullity | \(\operatorname{rank}+\operatorname{nullity}=\dim V\) |
| Injective | \(\ker T=\{\mathbf 0\}\); rank \(=\dim V\) |
| Surjective | \(\operatorname{Im}T=W\); rank \(=\dim W\) |
| Square operator | injective \(\iff\) surjective \(\iff\) invertible \(\iff\det\neq0\) |
| Matrix relation | \([T(\mathbf v)]_G=[T]_B^G[\mathbf v]_B\) |
| Change of basis | \([T]_{B'}=P^{-1}[T]_BP\) |
| Similarity invariants | det, tr, rank, char. poly, eigenvalues |
| \(\dim L(V,W)\) | \(\dim V\cdot\dim W=mn\) |
| Composition rank | \(\le\min(\operatorname{rank}S,\operatorname{rank}T)\) |
| Idempotent \(T^2=T\) | \(V=\ker T\oplus\operatorname{Im}T\) |
| Nilpotent \(T^2=0\) | \(\operatorname{Im}T\subseteq\ker T\) |
| Nilpotent on \(n\)-dim | \(T^n=0\); not invertible |
🎯 Ready to begin?
50 MCQs on Linear Transformations, Matrix Representation, Change of Basis and Rank–Nullity. One question at a time; full explanation, per-option analysis and TRB tip after each answer.
Difficulty blocks: Q1–10 Foundation · Q11–20 Definitions · Q21–30 Theorems · Q31–40 Numerical & Shortcuts · Q41–50 Advanced.
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