MAST10007 Linear Algebra
The Complete Exam Bible · S2 2026MAST10007 Overview
- First-year undergraduate
- Semester 2, 2026
- University of Melbourne
- Four assessment components
MAST10007 Linear Algebra is a first-year University of Melbourne mathematics subject that develops a single connected language for systems, matrices, vectors, abstract vector spaces and linear transformations.
- Assessed by 60% final · 10% mid-semester · 15% MATLAB · 15% assignments
- Final material One double-sided A4 notes sheet is the only permitted material
- Core method Represent → transform → interpret → verify
- MATLAB scope Introductory workflows grounded in the supplied Labs 1–2
What MAST10007 covers
MAST10007 moves from concrete systems and matrices into vector spaces, transformations, eigenstructure, orthogonality, least squares, SVD and complex linear algebra. This fourteen-chapter map follows that dependency chain while integrating the exact 2026 assessment conditions and the supported introductory MATLAB workflows.
Systems, Augmented Matrices and Row Operations
Linear systems · coefficient and augmented matrices · elementary row operations · elimination02Echelon Form, Consistency and Parametric Solutions
Pivot positions · echelon and reduced echelon form · free variables · solution sets03Matrix Algebra, Inverses, Rank and Determinants
Matrix products · inverses · elementary matrices · rank · determinant tests04Vectors, Dot and Cross Products, Lines and Planes
Vector geometry · orthogonality · cross products · parametric lines · plane equations05Vector Spaces and Subspaces
Axioms · standard examples · polynomial and matrix spaces · subspace test06Span and Linear Independence
Linear combinations · generating sets · dependence relations · geometric interpretation07Basis, Dimension, Coordinates and Fundamental Spaces
Bases · coordinate vectors · row, column and null spaces · pivot information08Linear Transformations and Standard Matrices
Linearity · matrix representations · composition · geometric transformations09Image, Kernel, Rank–Nullity and Change of Basis
Range and null space · one-to-one/onto · dimension theorem · coordinate conversion10Eigenvalues, Eigenvectors and Diagonalisation
Characteristic polynomials · eigenspaces · multiplicity · real and complex diagonalisation11Matrix Powers and Markov Chains
Powers via diagonalisation · transition matrices · stationary states · long-run behaviour12Inner Products, Projections and Gram–Schmidt
Inner-product geometry · orthogonality · projection · orthonormal bases13Least Squares, Orthogonal Matrices, Symmetry and SVD
Normal equations · orthogonal transformations · spectral theorem · singular values14Complex Linear Algebra, MATLAB and Exam Synthesis
Conjugate transpose · Hermitian and unitary matrices · supported MATLAB workflows · assessment strategyThe course begins with row reduction and matrix algebra, then uses span, independence, basis, dimension, image and kernel to explain what computations mean. Its second movement studies eigenvalues and diagonalisation, matrix powers and Markov chains, inner products and Gram–Schmidt, least squares, orthogonal and symmetric matrices, singular value decomposition, and complex Hermitian/unitary structure.
The 2026 Semester 2 assessment has four components: a three-hour final written examination worth 60%, a 45-minute open-book online mid-semester test in Week 6 worth 10%, a 45-minute restricted-material MATLAB test in Week 12 worth 15%, and four equally weighted written assignments worth 15% total. Current LMS and subject information control pass conditions and the final sitting.
The final permits one double-sided A4 notes sheet as its only material. The subject rewards exact calculation together with justification, notation and interpretation, so this guide pairs every method with a check and uses only newly authored examples. A reliable study cycle moves from definition to computation, geometric meaning and verification.
Row operations are checked against the original equations; bases are checked for both independence and span; eigenvectors are substituted back into the matrix equation; projections are tested by orthogonality; and least-squares results are read as a best approximation rather than an exact solution. MATLAB supports calculation and diagnosis, but the written argument still needs dimensions, conditions and a clear conclusion.
The chapter sequence therefore revisits the same invariants in several representations so students can recognise when two procedures are expressing one underlying structure.
How MAST10007 is assessed
| Component | Weight | Format |
|---|---|---|
| Final written examination | 60% | Three hours in the end-of-semester examination period; the only permitted material is one A4 notes sheet with writing on front and back. |
| Online mid-semester test | 10% | 45 minutes in the allocated computer-lab class in Week 6; open book; conducted through WebWork. |
| MATLAB test | 15% | 45 minutes in the allocated computer-lab class in Week 12; conducted through WebWork. Only writing materials for rough work; no calculators, phones, dictionaries, lecture notes, or other written/printed material. |
| Four written assignments | 15% total | Four equally weighted handwritten assignments submitted as a single PDF in Canvas; listed deadlines are 11:59 pm Sundays 16 Aug, 30 Aug, 20 Sep and 11 Oct. |
Change of basis with a direction check
- setupReconstruct v=2(1,0)+3(1,1)=(5,3).
- methodWrite (5,3)=a(1,1)+b(0,1)=(a,a+b).
- solveSolve a=5 and a+b=3, giving b=-2.
- checkTherefore [v]_C=(5,-2); reconstruction gives (5,3), confirming direction.
Key terms
- Pivot
- The leading position used to organise elimination and identify basic variables or columns.
- Basis
- An ordered independent spanning list that gives every vector unique coordinates.
- Kernel
- The inputs a linear transformation sends to zero.
- Eigenvector
- A nonzero vector whose direction is preserved by a linear map, up to scalar multiplication.
- Orthogonal projection
- The closest vector in a target subspace, characterised by an orthogonal residual.
- Singular value decomposition
- A factorisation A=UΣV^T that expresses a real matrix as orthogonal input directions, nonnegative scaling, and orthogonal output directions.
MAST10007 FAQ
How is MAST10007 assessed in Semester 2 2026?
The published split is 60% final written examination, 10% online mid-semester test, 15% MATLAB test and 15% across four equally weighted written assignments. The mid-semester is open book; the MATLAB test permits only writing materials for rough work; the final permits one double-sided A4 notes sheet as its only material.
Does MAST10007 publish a hurdle for this offering?
The available 2026 subject material does not state whether a hurdle applies. Check the current Canvas page and official University information for any later update rather than inferring a rule from silence.
What should go on the MAST10007 notes sheet?
Prioritise decision rules and assumption checks: solution classification, invertibility equivalences, subspace/basis tests, rank–nullity, coordinate direction, diagonalisation criteria, projection and Gram–Schmidt, normal equations, SVD, and conjugate-transpose identities. Keep one tiny checked example per error-prone method.
Can AI help me study MAST10007?
Yes. Ask Sia for a fresh example, a step-by-step explanation, or feedback on your reasoning. It should support learning rather than complete graded work, and University of Melbourne academic-integrity rules apply.
Where should I get current exam details?
Use the current MAST10007 Canvas site and your official personal timetable. The available material does not publish the final date, question count, current reading time, total marks, or whether a hurdle applies.
How to study for the exam
Follow the dependency chain. First make row reduction, consistency and matrix arithmetic automatic. Then connect pivots to span, independence, basis, coordinates, image and kernel. Only then layer in eigenvalues, diagonalisation and matrix powers. Finish with orthogonality, least squares, SVD and complex adjoints. Each week, reproduce one completed example on fresh numbers and perform a separate check.
Use the four assignments to practise complete explanations, the open-book mid-semester test to practise fast navigation, and Labs 1–2 to rehearse the supported MATLAB syntax. For the 60% final, compress decision rules and assumptions onto the one permitted double-sided A4 sheet, then practise retrieving methods without looking at it.
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