MTH2021 Linear Algebra with Applications
The Complete Exam Bible · S1 2027MTH2021 Overview
- Monash University
- S1 2027
- 13 chapters
- Mathematics
MTH2021 Linear Algebra with Applications is Monash University's second-year mathematics unit — dual-coded with the advanced MTH2025 and taught from Tim Garoni's printed seminar notes (with Anton & Rorres as the recommended text) — and it is the unit where, in the coordinator's own framing, rigorous mathematical proof takes centre stage.
- Assessed by Final Exam 50% · Test 2 20% · Twelve Weekly Quizzes 12% · Test 1 10% · Seminar Polls 4% · Applied-Class Participation 4%
- Key terms Reduced row-echelon form (RREF), Determinant, Basis and dimension, Rank–Nullity Theorem
- How to prepare Treat MTH2021 as two habits running in parallel: drill the recurring computations and rehearse the proofs.
- Most asked Where can I find past exam papers / practice for MTH2021?
How MTH2021 is assessed
| Component | Weight | Format |
|---|---|---|
| Final Exam | 50% | 190-minute invigilated on-campus eExam, including a proof-heavy handwritten section; exam period |
| Test 2 | 20% | Week 9; covers Weeks 3-7 |
| Twelve Weekly Quizzes | 12% | One per week (1% each); online Moodle, 5 questions each (True/False, MCQ, numeric) |
| Test 1 | 10% | Week 4; closed book; covers Weeks 1-2 |
| Seminar Polls | 4% | Poll Everywhere throughout the semester (graded by % correct in bands) |
| Applied-Class Participation | 4% | Weekly applied classes (Weeks 2-12); graded by number of classes actively participated in |
What MTH2021 covers
MTH2021 Linear Algebra with Applications is assessed through weekly Moodle quizzes, two in-semester tests (Test 1 in Week 4 on Weeks 1–2, Test 2 in Week 9 on Weeks 3–7) and a 190-minute invigilated eExam worth 50% that puts rigorous proof — not just computation — at centre stage.
This thirteen-chapter map follows the teaching schedule from Gaussian elimination through vector spaces, inner products and eigenvalues to the singular value decomposition. Use it to see how each week's definitions and theorems build toward the proof-heavy final.
Linear Systems & Gaussian Elimination
Augmented matrix · row-echelon & RREF · Gauss-Jordan · consistency (Week 1)Week 1 of Monash MTH2021 opens with systems of linear equations written as an augmented matrix [A | b], the three elementary row operations, and the march to row-echelon and reduced-row-echelon form via Gaussian and Gauss–Jordan elimination.
- Linear equation a₁x₁ + … + aₙxₙ = b; matrix form Ax = b and the augmented matrix [A | b]
- Elementary row operations: Rᵢ → cRᵢ (c ≠ 0), Rᵢ ↔ Rⱼ, Rᵢ → Rᵢ + cRⱼ; row equivalence A ~ B
- Row-echelon form vs reduced-row-echelon form (RREF); pivots unique in number and position, RREF unique
- Gaussian elimination (to echelon + back-substitution) and Gauss–Jordan elimination (to RREF)
- Leading vs free variables; free variables ↔ non-pivot columns
- Consistency: r = n pivots ⇒ 0 or 1 solution; r < n ⇒ 0 or ∞; if a system has >1 solution it has infinitely many
Matrix Algebra, Inverses & Elementary Matrices
Matrix operations · the inverse & inversion algorithm · elementary matrices · equivalent statements (Weeks 1-2)Weeks 1–2 build the algebra of matrices — addition, the product (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ, transpose, trace — and then the inverse, computed by the inversion algorithm that row-reduces [A | I] to [I | A⁻¹].
- Matrix operations and laws: (A+B), kA, product (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ; multiplication is associative but NOT commutative
- Transpose (Aᵀ)ⱼᵢ = (A)ᵢⱼ, symmetric matrices Aᵀ = A, trace tr(A) = Σ aᵢᵢ with tr(AB) = tr(BA)
- Ax as a linear combination of the columns of A (Ax = x₁a₁ + … + xₙaₙ)
- The inverse: A invertible ⇔ ∃B, AB = BA = I; uniqueness; (AB)⁻¹ = B⁻¹A⁻¹, (Aᵀ)⁻¹ = (A⁻¹)ᵀ
- Elementary matrices E = ρ(I); ρ(A) = EA; every elementary matrix is invertible
- The inversion algorithm: row-reduce [A | I] → [I | A⁻¹]; a zero row on the left ⇒ singular
Determinants
Cofactor / Laplace expansion · properties · determinant of a product · Cramer's rule (Week 3)Week 3 introduces the determinant through recursive cofactor (Laplace) expansion, the effect of row operations on its value, and the headline properties det(AB) = det(A)det(B), det(Aᵀ) = det(A) and det(A⁻¹) = 1/det(A).
- Recursive definition: det[a] = a, det of a 2×2 = ad − bc; minor Mᵢⱼ and cofactor Cᵢⱼ = (−1)^{i+j}Mᵢⱼ
- Cofactor (Laplace) expansion along ANY row or column gives the same value
- A row or column of zeros ⇒ det = 0; triangular matrix ⇒ det = product of diagonal entries
- Row-operation effects: scale a row by k ⇒ det × k; swap two rows ⇒ det × (−1); add a multiple of one row to another ⇒ det unchanged
- det(kA) = kⁿ det(A); det(EA) = det(E)det(A); det(AB) = det(A)det(B); det(A⁻¹) = 1/det(A)
- det(Aᵀ) = det(A) — expansion along a row equals expansion along the matching column
Vector Spaces & Subspaces
The axioms · the subspace test · examples (function/matrix/polynomial spaces) (Weeks 3-4)Weeks 3–4 make the leap from computation to structure: a vector space is a set with addition and scalar multiplication satisfying ten axioms, and a subspace is a nonempty subset closed under both operations.
- The ten vector-space axioms (closure, commutativity, associativity, zero, negatives, scalar-multiplication laws)
- Consequences: 0 is unique, each v has a unique −v, 0·u = 0, k·0 = 0, (−1)u = −u, ku = 0 ⇒ k = 0 or u = 0
- Standard examples: {0}, ℝⁿ, ℝ^∞, matrix space ℝ^{m×n}, polynomials Pₙ and P∞, function space C[0,1]
- The subspace test: nonempty W ⊆ V is a subspace ⇔ closed under addition and under scalar multiplication
- A subspace must contain the zero vector (a fast disqualifier)
- Subspace examples: lines/planes through the origin, symmetric matrices in ℝ^{2×2}, Pₙ ⊂ P∞
Span, Linear Independence, Basis & Dimension
Linear combinations · span · independence · basis · the dimension theorem (Week 4)Week 4 assembles the core structural toolkit: linear combinations and span, linear (in)dependence, a basis as a minimal spanning set, and dimension.
- Linear combination v = a₁w₁ + … + aᵣwᵣ; span(S) = all linear combinations = the smallest subspace containing S
- Linear independence: k₁v₁ + … + kᵣvᵣ = 0 ⇒ all kᵢ = 0; otherwise dependent
- A set is dependent ⇔ some vector is a linear combination of the others
- Basis = linearly independent + spanning; standard bases of ℝⁿ, Pₙ, ℝ^{2×2}
- In an n-vector basis space: more than n vectors ⇒ dependent; fewer than n ⇒ cannot span
- Dimension Theorem: every basis of a finite-dimensional V has the same size, dim(V)
Coordinates, Change of Basis & Direct Sums
Coordinate vectors [v]_B · change-of-basis matrix · direct sum V (+) W (Week 5)Week 5 turns a basis into a coordinate system: relative to a basis B, every vector has a unique coordinate vector [v]_B, and the change-of-basis matrix P converts coordinates between two bases via [v]_{B'} = P[v]_B. It also introduces the direct sum V ⊕ W.
- Unique expansion of v in a basis B; the coordinate vector [v]_B ∈ ℝⁿ
- Coordinates in polynomial and matrix spaces, e.g. [a₀ + a₁x + a₂x²] in {1, x, x²}
- Change-of-basis matrix P_{B'←B} whose columns are the old basis vectors' coordinates in the new basis
- The transformation law [v]_{B'} = P_{B'←B} [v]_B
- P_{B'←B} is invertible and (P_{B'←B})⁻¹ = P_{B←B'}
- Sum of subspaces V + W = {v + w}; it is a direct sum V ⊕ W when V ∩ W = {0}
Fundamental Subspaces, Fields & Coding Theory
Row/column space · kernel & cokernel · vector spaces over a field · Hamming (7,4) codes (Week 6)Week 6 attaches four subspaces to every matrix — the row space, column space, kernel (null space) and cokernel — and reads their dimensions off the rank.
- The four fundamental subspaces: row(A), col(A), ker(A) = {x : Ax = 0}, coker(A) = ker(Aᵀ)
- Rank r = number of pivots = dim(row(A)) = dim(col(A)); ker and coker are subspaces
- Dimensions: dim row = dim col = r; dim ker(A) = n − r; dim coker(A) = m − r
- Ax = b is consistent ⇔ b ∈ col(A)
- Fields F (axioms F1–F6); examples ℚ, ℝ, ℂ, Z_p for prime p; vector spaces over a field, Fⁿ
- Modular arithmetic in Z_p and binary vector spaces over Z₂
Linear Transformations & the Rank-Nullity Theorem
Kernel & range · rank-nullity · injective/surjective · matrix of a map · isomorphism (Weeks 6-7)Weeks 6–7 study linear transformations T : V → W through their kernel and range, the Rank–Nullity Theorem rank(T) + nullity(T) = dim(V), and the notions of injective, surjective and isomorphism.
- Linear map: T(u + kv) = T(u) + kT(v); consequence T(0) = 0; matrix transformations T_A(x) = Ax are linear
- Kernel ker(T) = {v : T(v) = 0} and range ran(T); both are subspaces; for T_A, ker(T_A) = ker(A), ran(T_A) = col(A)
- rank(T) = dim ran(T); nullity(T) = dim ker(T)
- Rank–Nullity Theorem: rank(T) + nullity(T) = n = dim(V)
- T injective ⇔ ker(T) = {0}; for equal finite dimensions, injective ⇔ surjective
- Isomorphism = bijective linear map; V ≅ W; dim(U) = dim(V) ⇒ U ≅ V
Inner Product Spaces
Inner-product axioms · norm & distance · Cauchy-Schwarz · orthogonality & complements (Week 8)Week 8 adds geometry to vector spaces through an inner product ⟨u, v⟩ satisfying symmetry, linearity and positive definiteness.
- Inner-product axioms: symmetry ⟨u,v⟩ = ⟨v,u⟩, linearity ⟨u, v + kw⟩ = ⟨u,v⟩ + k⟨u,w⟩, positive definiteness ⟨v,v⟩ ≥ 0 with equality ⇔ v = 0
- Examples: the dot product on ℝⁿ; weighted ⟨u,v⟩ = Σ wᵢuᵢvᵢ (wᵢ > 0); integral ⟨f,g⟩ = ∫ f g on function spaces
- Induced norm ‖v‖ = √⟨v,v⟩ and distance d(u,v) = ‖u − v‖
- Cauchy–Schwarz inequality: |⟨u,v⟩| ≤ ‖u‖‖v‖
- Triangle inequality ‖u + v‖ ≤ ‖u‖ + ‖v‖ and the norm/metric properties
- Angle θ = arccos(⟨u,v⟩ / (‖u‖‖v‖)); orthogonality ⟨u,v⟩ = 0
Gram-Schmidt, Projection & Least Squares
Orthonormal bases via Gram-Schmidt · the projection theorem · normal equations & best fit (Weeks 9-10)Weeks 9–10 turn any basis into an orthonormal one via the Gram–Schmidt process, project vectors onto subspaces with the projection theorem, and solve overdetermined systems by least squares.
- Orthogonal and orthonormal sets; an orthogonal set of nonzero vectors is automatically linearly independent
- Gram–Schmidt process: v₁ = u₁, then vₖ = uₖ − Σ_{i<k} (⟨uₖ, vᵢ⟩ / ‖vᵢ‖²) vᵢ; normalise for an orthonormal basis
- Projection Theorem: every u = w + w^⊥ with projᵥᵥ u = Σⱼ ⟨u, vⱼ⟩ vⱼ for an orthonormal basis of W
- V = W ⊕ W^⊥; every finite-dimensional inner product space has an orthonormal basis
- Fundamental Theorem of Linear Algebra: row(A)^⊥ = ker(A); ℝⁿ = row(A) ⊕ ker(A), ℝᵐ = col(A) ⊕ coker(A)
- Closest-point / least-squares problem: minimise ‖b − Ax‖; the error vector is b − Ax
Eigenvalues, Eigenvectors & Diagonalisation
Characteristic polynomial · eigenspaces & multiplicities · A = PDP^-1 (Weeks 10-11)Weeks 10–11 solve Ax = λx: eigenvalues from the characteristic equation det(λI − A) = 0, eigenspaces as kernels of λI − A, algebraic vs geometric multiplicity, and diagonalisation A = PDP⁻¹ when A has n independent eigenvectors.
- Eigenvector x ≠ 0 with Ax = λx; scalar multiples cx are also eigenvectors
- Characteristic equation det(λI − A) = 0; the characteristic polynomial is monic of degree n, so ≤ n distinct eigenvalues
- Triangular matrices have their diagonal entries as eigenvalues; A invertible ⇔ 0 is not an eigenvalue
- Eigenspace = ker(λI − A); λ eigenvalue of A ⇒ λᵏ eigenvalue of Aᵏ (same eigenvector)
- Algebraic multiplicity (power in the characteristic polynomial) vs geometric multiplicity (dim of eigenspace); geometric ≤ algebraic
- Similar matrices B = P⁻¹AP share determinant, characteristic polynomial and geometric multiplicities
Orthogonal Diagonalisation & the Spectral Theorem
Orthogonal matrices · symmetric matrices · A = QDQ^T · spectral decomposition (Week 11)Week 11 specialises diagonalisation to symmetric matrices.
- Orthogonal matrix: A⁻¹ = Aᵀ; rows (and columns) form an orthonormal basis
- Orthogonal matrices preserve length and dot product (‖Ax‖ = ‖x‖, Ax·Ay = x·y); det = ±1; products of orthogonal matrices are orthogonal
- Change-of-basis between orthonormal bases is an orthogonal matrix; rotations R_θ and reflections are examples
- Orthogonally diagonalisable: A = PDPᵀ with P orthogonal
- Spectral Theorem: A is orthogonally diagonalisable ⇔ A is symmetric ⇔ A has an orthonormal set of n eigenvectors
- Symmetric matrices: eigenvectors for distinct eigenvalues are automatically orthogonal
Singular Value Decomposition
Singular values · A = U S V^T · the pseudoinverse · minimal-length least squares (Week 12)Week 12 extends diagonalisation to every matrix — rectangular or singular — via the singular value decomposition A = UΣVᵀ. The singular values σᵢ = √(eigenvalues of AᵀA) sit on Σ; the columns of V are eigenvectors of AᵀA and uᵢ = Avᵢ/σᵢ build U.
- AᵀA is symmetric with nonnegative eigenvalues; singular values σᵢ = √λᵢ of AᵀA
- A matrix of rank r has exactly r positive singular values
- SVD A = UΣVᵀ: U (m×m orthogonal), Σ (m×n nonnegative diagonal), V (n×n orthogonal)
- Columns of V = orthonormal eigenvectors of AᵀA (ordered by decreasing σ); uᵢ = Avᵢ/σᵢ for i ≤ r; extend U with a coker(A) basis
- The singular vectors give orthonormal bases of the four fundamental subspaces
- Singular value expansion A = σ₁u₁v₁ᵀ + … + σᵣuᵣvᵣᵀ (reduced SVD)
The Monash Handbook frames MTH2021 in two banded halves: a continuous 'Demonstration' component worth 50% (seminar polls 4%, applied-class participation 4%, twelve weekly Moodle quizzes 12%, Test 1 10% in Week 4 covering Weeks 1–2, and Test 2 20% in Week 9 covering Weeks 3–7) and a single Final Exam worth 50%.
That final is a 190-minute invigilated on-campus eExam with a proof-heavy handwritten section, so marks reward both a clean computation and a correctly argued proof. There is no hurdle in the current Monash materials — confirm on Moodle whether any pass-the-exam rule applies.
The content runs from Gaussian elimination and determinants through abstract vector spaces, inner products and eigenvalues to the singular value decomposition, and the MTH2021 result feeds the Weighted Average Mark (WAM) that later Monash mathematics units build on. Generative AI cannot be used for any assessment in this unit.
Worked example · free
Least-squares line of best fit via the normal equations
- +1Set up the design matrix and data vector. With rows [1, xᵢ], M = [[1,0],[1,1],[1,3],[1,6]] and y = (2, 3, 7, 12)ᵀ; the unknowns are v = (a, b)ᵀ. The least-squares fit solves MᵀM v = Mᵀy, which is always consistent.
- +1Form the Gram matrix MᵀM. With n = 4, Σx = 0+1+3+6 = 10 and Σx² = 0+1+9+36 = 46, MᵀM = [[4, 10],[10, 46]].
- +1Form the right-hand side Mᵀy. Σy = 2+3+7+12 = 24 and Σxy = 0·2 + 1·3 + 3·7 + 6·12 = 0+3+21+72 = 96, so Mᵀy = (24, 96)ᵀ.
- +1Solve [[4,10],[10,46]] (a, b)ᵀ = (24, 96)ᵀ. The determinant is 4·46 − 10·10 = 184 − 100 = 84. By Cramer's rule a = (24·46 − 10·96)/84 = (1104 − 960)/84 = 144/84 = 12/7, and b = (4·96 − 10·24)/84 = (384 − 240)/84 = 144/84 = 12/7.
- +1State the line and sanity-check. y = 12/7 + (12/7)x ≈ 1.71 + 1.71x. At x = 6 it predicts 12/7·7 = 12, matching the point (6, 12) exactly, and the fitted values (1.71, 3.43, 6.86, 12.0) sit close to the data — a good fit.
Key terms
- Reduced row-echelon form (RREF)
- The unique echelon form of a matrix in which every pivot is 1 and is the only nonzero entry in its column. Gauss–Jordan elimination reaches it; the number and position of pivots (hence the rank) are invariant.
- Determinant
- A scalar det(A) attached to a square matrix, computed by cofactor (Laplace) expansion along any row or column. A is invertible iff det(A) ≠ 0, and det(AB) = det(A)det(B).
- Basis and dimension
- A basis is a linearly independent spanning set; every basis of a finite-dimensional space has the same size, which is the dimension. dim(ℝⁿ) = n, dim(Pₙ) = n+1, dim(ℝ^{m×n}) = mn.
- Rank–Nullity Theorem
- For a linear map T on an n-dimensional domain, rank(T) + nullity(T) = n — the dimension of the range plus the dimension of the kernel equals the dimension of the domain.
- Eigenvalue and eigenvector
- A nonzero vector x with Ax = λx; λ is the eigenvalue, found from the characteristic equation det(λI − A) = 0. A is diagonalisable, A = PDP⁻¹, exactly when it has n linearly independent eigenvectors.
- Singular value decomposition (SVD)
- Any m×n matrix factors as A = UΣVᵀ with U, V orthogonal and Σ diagonal of nonnegative singular values σᵢ = √(eigenvalues of AᵀA). It underlies the pseudoinverse A⁺ and the minimal-length least-squares solution.
MTH2021 FAQ
Is MTH2021 hard?
MTH2021 is a step up because it is the unit where rigorous proof takes centre stage, so it is harder than the computation-focused first-year linear algebra in MTH1030 or ENG1005.
The computations (row reduction, determinants, eigenvalues, Gram–Schmidt, SVD) are drillable, but the marks in the 190-minute final increasingly reward proving why a result holds — that a set is a subspace, that a map is an isomorphism, that a matrix is orthogonally diagonalisable.
Students who keep up with the weekly quizzes and rehearse the proofs, rather than only computing, tend to find it manageable and land in the HD/D bands.
Can AI help me with MTH2021?
Yes — as a step-by-step explainer, not an answer machine. Sia is an AI tutor trained on how MTH2021 is actually taught and assessed: it can walk you through a Gaussian-elimination consistency argument, explain why only the space of eigenvectors makes A diagonalisable, or unpack a Gram–Schmidt or SVD calculation one line at a time, and it checks your reasoning as you go.
It does not do graded assessment for you — Monash University academic-integrity rules apply, and generative AI cannot be used for any assessment in this unit — so use it to understand the method and rehearse proofs, then write your own.
Where can I find past exam papers / practice for MTH2021?
Start on Moodle: the unit posts sample final exams (with solutions for the handwritten section), sample tests, twelve weekly problem sets with full worked solutions, and the quizzes — that is the closest match to the real exam's style.
This guide also includes a re-authored practice exam that mirrors the final's shape (a computation half plus a proof-heavy handwritten half) with fresh numbers, and you can ask Sia to generate extra practice in the same style and explain each step. Treat any third-party 'model answers' with caution and confirm what is officially provided on Moodle and the unit guide.
Does MTH2021 have a hurdle?
No hurdle appears in the current Monash Handbook entry or the Moodle materials — both list the two assessment bands (continuous 'Demonstration' 50% and Final Exam 50%) with no hurdle attached.
Your final mark is simply the weighted total across seminar polls (4%), applied-class participation (4%), twelve quizzes (12%), Test 1 (10%), Test 2 (20%) and the exam (50%), mapped to the Monash grade bands (HD ≥ 80, then D, C, P; N is a fail). Still, confirm on Moodle whether any pass-the-exam rule applies to your cohort before you rely on this.
Is the MTH2021 final open- or closed-book?
The final is a 190-minute invigilated on-campus eExam with a proof-heavy handwritten section, but the available unit materials do not state whether it is open- or closed-book or which materials are permitted — so do not assume either way.
(Test 1 in Week 4 is explicitly closed book, but that does not settle the exam.) Check the exam cover sheet, Moodle and the Monash exam timetable for the exact date — around June 2027 in the Semester 1 exam period — time, room and permitted-materials rules before the day.
Questions students ask in each MTH2021 chapter
- How do I read off unique / infinite / no solution from an echelon form?
- Look at the pivots after elimination. A row (0 … 0 | nonzero) means no solution. Otherwise, if every variable column has a pivot the solution is unique; if some variable column has no pivot, that variable is free and there are infinitely many solutions. The number of free variables equals n − r, where r is the number of pivots. Chapter 01 · Linear Systems & Gaussian Elimination
- How does the inversion algorithm actually find A⁻¹?
- Row-reducing [A | I] to RREF applies a sequence of elementary matrices E_k…E₁ that turns A into I. So E_k…E₁·A = I, which means E_k…E₁ = A⁻¹. The same operations acting on the I block on the right therefore assemble A⁻¹ directly. If the left block cannot reach I (a zero row appears), A is singular. Chapter 02 · Matrix Algebra, Inverses & Elementary Matrices
- Which row or column should I expand along?
- Whichever has the most zeros, because each zero entry removes an entire cofactor term. Cofactor expansion gives the same determinant along any row or column (that is a theorem), so you are free to choose the easiest one. For larger matrices it is often faster to row-reduce to triangular form and multiply the diagonal, tracking the det factors from each operation. Chapter 03 · Determinants
- What exactly do I check in the subspace test?
- Three things: the set is nonempty (show 0 is in it), it is closed under addition (u, v in W ⇒ u + v in W), and it is closed under scalar multiplication (u in W, scalar k ⇒ ku in W). If all three hold, W is a subspace. If any fails, it is not — and failing the 0 check is the quickest disqualifier. Chapter 04 · Vector Spaces & Subspaces
- How do I test whether vectors are linearly independent?
- Form a matrix from the vectors and row-reduce. If the number of pivots equals the number of vectors, they are independent; if there are fewer pivots, they are dependent. For n vectors in ℝⁿ you can shortcut with the determinant: nonzero ⇒ independent, zero ⇒ dependent. Chapter 05 · Span, Linear Independence, Basis & Dimension
- What is a coordinate vector, really?
- Once you fix a basis B = {v₁, …, vₙ}, every vector v has one and only one expansion v = c₁v₁ + … + cₙvₙ, and the coefficients (c₁, …, cₙ) are its coordinate vector [v]_B. This turns abstract vectors — polynomials, matrices, anything — into ordinary columns in ℝⁿ, which is why the whole computational machinery transfers. Chapter 06 · Coordinates, Change of Basis & Direct Sums
- What are the four fundamental subspaces and how do I find them?
- They are the row space and kernel (living in ℝⁿ) and the column space and cokernel (living in ℝᵐ). Row-reduce A once: the pivots give the rank r; nonzero RREF rows span row(A); the original columns in pivot positions span col(A); free variables give ker(A); and coker(A) = ker(Aᵀ) has dimension m − r. Chapter 07 · Fundamental Subspaces, Fields & Coding Theory
- How do I build the matrix of a linear map?
- Apply T to each vector of the domain basis B, express each image in the codomain basis B', and use those coordinate vectors as the columns of [T]_{B',B}. The resulting matrix satisfies [T][x]_B = [T(x)]_{B'}, so once you have it, evaluating T is just matrix multiplication. Chapter 08 · Linear Transformations & the Rank-Nullity Theorem
- What makes something an inner product?
- Three axioms: symmetry ⟨u,v⟩ = ⟨v,u⟩, linearity in one slot, and positive definiteness (⟨v,v⟩ ≥ 0, equal to 0 only when v = 0). The dot product is the model, but weighted sums Σwᵢuᵢvᵢ (with positive weights) and integrals ∫fg on function spaces also qualify, and each gives its own notion of length and angle. Chapter 09 · Inner Product Spaces
- Do I normalise during Gram–Schmidt or at the end?
- At the end. Run the subtraction step vₖ = uₖ − Σ (⟨uₖ, vᵢ⟩/‖vᵢ‖²)vᵢ with the un-normalised orthogonal vectors, checking orthogonality after each one, and only divide by the norms once all the vₖ are found. Normalising midway is allowed but just multiplies the arithmetic. Chapter 10 · Gram-Schmidt, Projection & Least Squares
- How do I find eigenvalues and eigenvectors?
- Solve the characteristic equation det(λI − A) = 0 for the eigenvalues, then for each λ find the eigenspace by computing ker(λI − A) — the nonzero solutions of (A − λI)x = 0. Scalar multiples of an eigenvector are again eigenvectors, so you report a basis of each eigenspace. Chapter 11 · Eigenvalues, Eigenvectors & Diagonalisation
- What makes a matrix orthogonally diagonalisable?
- The Spectral Theorem gives a clean answer: a real matrix is orthogonally diagonalisable (A = QDQᵀ with Q orthogonal) if and only if it is symmetric. So you never have to hunt — check symmetry, and if Aᵀ = A you are guaranteed an orthonormal eigenbasis. Chapter 12 · Orthogonal Diagonalisation & the Spectral Theorem
- How do I actually compute an SVD?
- Start from AᵀA: find its orthonormal eigenvectors and order them by decreasing eigenvalue to form V, and take σᵢ = √λᵢ for Σ. Then set uᵢ = Avᵢ/σᵢ for each positive singular value, and if A is not square (or is rank-deficient) complete U with an orthonormal basis of ker(Aᵀ). Assemble A = UΣVᵀ. Chapter 13 · Singular Value Decomposition
How to study for the exam
Treat MTH2021 as two habits running in parallel: drill the recurring computations and rehearse the proofs. Each week, do that week's problem set end to end and one full quiz-style self-test, because the twelve Moodle quizzes (12%) track the week's topic and Test 1 (Weeks 1–2) and Test 2 (Weeks 3–7) re-examine the same material.
For the proof-heavy 50% final, do not only compute: practise writing out the subspace test, the rank–nullity argument, the diagonalisation and spectral-theorem proofs, and the union-is-not-a-subspace counterexample until you can produce them cold, since the handwritten section is where those marks live.
Use SWOTVAC to consolidate breadth across all thirteen chapters rather than to meet the material for the first time, and confirm the exam date, room and permitted materials on Moodle. When a step won't click, ask Sia to re-explain that single line a different way and to set a fresh practice question in the same style — it teaches the method and checks your reasoning, and it never substitutes for your own graded work.
Your AI Mathematics tutor for MTH2021
Stuck on a hard MTH2021 question? Sia is AskSia’s AI Mathematics tutor — ask any MTH2021 Linear Algebra with Applications question and get a clear, step-by-step explanation grounded in how the course is actually taught and assessed. Read this whole study guide free, then take your hardest questions to Sia.
