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 you get in the full MAST10007 Exam Bible
- 14 chapters, 160 pages — every examinable topic, week by week
- 48 original practice questions, most with model answers in the Final written examination format
- 14 worked examples, solved the way MAST10007 marks
- 28 computed figures, redrawn from the course’s own numbers
- Exam-morning plan for the Final written examination (60%) — what to revise last, where marks are lost
Free on this page: assessment table, chapter map, one worked example, key terms, FAQ. The rest is in the full exam bible.
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. |
What other students are asking about MAST10007
- Where can I find lecture notes or study guides for MAST10007 at UniMelb?
- The MAST10007 study guide in AskSia Library covers Linear Algebra in 14 chapters, with the assessment table, a chapter-by-chapter topic map, key terms and a worked example, free to read on this page.
- Is there a final exam in MAST10007, and how much is it worth?
- Yes. In the offering this AskSia Library guide is built from, Final written examination carries 60% of the MAST10007 mark; the full split is Final written examination 60%, Online mid-semester test 10%, MATLAB test 15%, Four written assignments 15% total.
- What does MAST10007 cover on echelon form, consistency and parametric solutions?
- In AskSia Library's MAST10007 guide, the Echelon Form, Consistency and Parametric Solutions chapter explains pivot positions · echelon and reduced echelon form · free variables · solution sets.
- What does MAST10007 cover on matrix algebra, inverses, rank and determinants?
- In AskSia Library's MAST10007 guide, the Matrix Algebra, Inverses, Rank and Determinants chapter explains matrix products · inverses · elementary matrices · rank · determinant tests.
- What does MAST10007 cover on vectors, dot and cross products, lines and planes?
- In AskSia Library's MAST10007 guide, the Vectors, Dot and Cross Products, Lines and Planes chapter explains vector geometry · orthogonality · cross products · parametric lines · plane equations.
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 · eliminationChapter 1 develops systems, augmented matrices and row operations for University of Melbourne MAST10007. A linear system is a collection of equations imposed at once.
- From equations to matrix form
- Coefficient versus augmented matrices
- The three legal row operations
- Forward elimination and pivot choice
- Back-substitution
- Elimination as reversible logic
Echelon Form, Consistency and Parametric Solutions
Pivot positions · echelon and reduced echelon form · free variables · solution setsChapter 2 develops echelon form, consistency and parametric solutions for University of Melbourne MAST10007. Echelon form is a structure, not one unique matrix: nonzero rows sit above zero rows and leading entries move right.
- Recognising echelon form
- Reduced echelon form
- Pivot and free columns
- Contradiction rows
- Unique solutions
- Infinitely many solutions
Matrix Algebra, Inverses, Rank and Determinants
Matrix products · inverses · elementary matrices · rank · determinant testsChapter 3 develops matrix algebra, inverses, rank and determinants for University of Melbourne MAST10007. Matrix multiplication records composition and is order-sensitive.
- Matrix dimensions and entries
- Addition and scalar multiplication
- Row-by-column products
- Identity and transpose
- Inverse matrices
- Elementary matrices
Vectors, Dot and Cross Products, Lines and Planes
Vector geometry · orthogonality · cross products · parametric lines · plane equationsChapter 4 develops vectors, dot and cross products, lines and planes for University of Melbourne MAST10007. Vectors carry magnitude and direction. The dot product measures alignment and gives orthogonality when it is zero.
- Vector arithmetic and length
- Dot product and angle
- Orthogonality
- Scalar and vector projection
- Cross product
- Area and orientation
Vector Spaces and Subspaces
Axioms · standard examples · polynomial and matrix spaces · subspace testChapter 5 develops vector spaces and subspaces for University of Melbourne MAST10007. A vector space is defined by operations and axioms, not by arrows.
- Why abstract vector spaces
- Vector-space axioms
- Zero and additive inverses
- Polynomial spaces
- Matrix and function spaces
- Subspace test
Span and Linear Independence
Linear combinations · generating sets · dependence relations · geometric interpretationChapter 6 develops span and linear independence for University of Melbourne MAST10007. The span of vectors is the set of every linear combination they can generate. Independence asks whether the zero vector has only the trivial representation.
- Linear combinations
- Span as all combinations
- Geometric pictures of span
- Testing membership in a span
- Independence definition
- Dependence relations
Basis, Dimension, Coordinates and Fundamental Spaces
Bases · coordinate vectors · row, column and null spaces · pivot informationChapter 7 develops basis, dimension, coordinates and fundamental spaces for University of Melbourne MAST10007. A basis is a coordinate system: it spans the space without redundancy, so every vector has one coordinate list.
- Basis as span plus independence
- Dimension
- Coordinates relative to a basis
- Extracting a column-space basis
- Row-space basis
- Null-space basis
Linear Transformations and Standard Matrices
Linearity · matrix representations · composition · geometric transformationsChapter 8 develops linear transformations and standard matrices for University of Melbourne MAST10007. A transformation is linear when it preserves vector addition and scalar multiplication.
- Transformation language
- The two linearity conditions
- Consequences of linearity
- Standard matrices
- Images of basis vectors
- Rotations and reflections
Image, Kernel, Rank–Nullity and Change of Basis
Range and null space · one-to-one/onto · dimension theorem · coordinate conversionChapter 9 develops image, kernel, rank–nullity and change of basis for University of Melbourne MAST10007. The kernel records inputs erased by a linear map; the image records outputs the map can actually reach.
- Kernel
- Image
- One-to-one and the kernel
- Onto and the image
- Rank–nullity
- Finding bases for image and kernel
Eigenvalues, Eigenvectors and Diagonalisation
Characteristic polynomials · eigenspaces · multiplicity · real and complex diagonalisationChapter 10 develops eigenvalues, eigenvectors and diagonalisation for University of Melbourne MAST10007. An eigenvector is a nonzero direction that a linear transformation only scales.
- Eigenvector meaning
- Characteristic equation
- Finding eigenspaces
- Algebraic multiplicity
- Geometric multiplicity
- Diagonalisation criterion
Matrix Powers and Markov Chains
Powers via diagonalisation · transition matrices · stationary states · long-run behaviourChapter 11 develops matrix powers and markov chains for University of Melbourne MAST10007. Diagonalisation turns repeated matrix multiplication into repeated scalar powers.
- Why matrix powers matter
- Powers from diagonal form
- Recursive linear systems
- Transition matrices
- Column versus row conventions
- Probability-vector checks
Inner Products, Projections and Gram–Schmidt
Inner-product geometry · orthogonality · projection · orthonormal basesChapter 12 develops inner products, projections and gram–schmidt for University of Melbourne MAST10007. An inner product generalises the dot product and creates length, angle, and orthogonality.
- Inner products
- Norm and distance
- Orthogonality
- Cauchy–Schwarz
- Projection onto a vector
- Projection onto a subspace
Least Squares, Orthogonal Matrices, Symmetry and SVD
Normal equations · orthogonal transformations · spectral theorem · singular valuesChapter 13 develops least squares, orthogonal matrices, symmetry and svd for University of Melbourne MAST10007. Least squares chooses Ax closest to b, so the residual b-Ax is orthogonal to the column space. That condition yields the normal equations.
- Inconsistent systems and best fit
- Orthogonal residuals
- Normal equations
- Projection viewpoint
- Orthogonal matrices
- Length and angle preservation
Complex Linear Algebra, MATLAB and Exam Synthesis
Conjugate transpose · Hermitian and unitary matrices · supported MATLAB workflows · assessment strategyChapter 14 develops complex linear algebra, matlab and exam synthesis for University of Melbourne MAST10007. Complex inner products use conjugation so that a vector has a real nonnegative squared norm.
- Complex vectors and conjugation
- Conjugate transpose
- Complex inner products
- Hermitian matrices
- Unitary matrices
- Real special cases
The 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.
MAST10007 cheatsheet for the final exam
AskSia publishes a free MAST10007 cheatsheet for the final exam, an A4 cheat sheet that follows the official permitted-materials rule for this assessment and condenses the topics in this guide. Open the MAST10007 cheatsheet.
Worked example · free
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.
Questions students ask in each MAST10007 chapter
- How do I check from equations to matrix form?
- Writing a system in matrix form fixes an order for the variables. A missing variable contributes a zero coefficient; changing that order halfway through changes the problem. The augmented divider separates coefficients from constants but does not shield the constants from row operations. R2←R2-2R1 gives (0,-3|-7), so y=7/3 and x=5/3. Chapter 01 · Systems, Augmented Matrices and Row Operations
- How do I check recognising echelon form?
- Row-echelon form requires zero rows at the bottom and every leading entry strictly to the right of the leading entry above it. Pivot entries need not be one. Many echelon forms represent the same system, so the form is recognised by structure rather than a unique appearance. Consistent with one free variable, hence infinitely many solutions. Chapter 02 · Echelon Form, Consistency and Parametric Solutions
- How do I check matrix dimensions and entries?
- An m×n matrix has m rows and n columns, and its (i,j) entry sits at their intersection. Dimensions are part of the object. Before any operation, write the dimensions beside each factor; this catches illegal sums and products before arithmetic begins. The product is [[11],[4]]. Chapter 03 · Matrix Algebra, Inverses, Rank and Determinants
- How do I check vector arithmetic and length?
- Vector addition combines components and scalar multiplication changes magnitude and possibly direction. The Euclidean norm is the square root of the sum of squared components. A unit vector is obtained only from a nonzero vector by dividing by its norm. Yes: their dot product is 2-2+0=0. Chapter 04 · Vectors, Dot and Cross Products, Lines and Planes
- How do I check why abstract vector spaces?
- Abstract vector spaces retain addition and scalar multiplication while changing what a vector looks like. Coordinates, polynomials, matrices, and functions can all qualify. The axioms guarantee that linear combinations behave consistently across these settings. No; (0,0) does not satisfy the equation. Chapter 05 · Vector Spaces and Subspaces
- How do I check linear combinations?
- A linear combination c1v1+⋯+ckvk uses scalar coefficients to build a vector. The order of a list does not change its span, but it will later affect coordinate vectors. Write the ambient space so coefficient and dimension claims have a home. No; the third vector is the sum of the first two. Chapter 06 · Span and Linear Independence
- How do I check basis as span plus independence?
- A basis combines two properties: spanning ensures every vector can be expressed, and independence makes that expression unique. Omitting either property leaves a generating list or an independent list, not a coordinate system. Three, by nullity n-r=5-2. Chapter 07 · Basis, Dimension, Coordinates and Fundamental Spaces
- How do I check transformation language?
- A transformation assigns an output to each input. Its domain and codomain are part of its definition. The range may be smaller than the codomain, so those words must not be used interchangeably. No, because T(0,0)=(1,0), not zero. Chapter 08 · Linear Transformations and Standard Matrices
- How do I check kernel?
- The kernel is every input sent to zero. It is found by solving Ax=0 and lives in the domain. A nonzero kernel vector records a lost direction and proves the transformation is not one-to-one. Exactly when ker(T)={0}. Chapter 09 · Image, Kernel, Rank–Nullity and Change of Basis
- How do I check eigenvector meaning?
- An eigenvector is a nonzero vector whose image remains on its span: Av=λv. The scalar λ may reverse direction, shrink, stretch, or annihilate the vector. Zero is excluded because it satisfies the equation for every λ and carries no direction. Its characteristic equation λ²+1=0 has no real eigenvalues. Chapter 10 · Eigenvalues, Eigenvectors and Diagonalisation
- How do I check why matrix powers matter?
- Directly multiplying A many times is expensive and hides structure. If A=PDP^{-1}, middle factors cancel and A^k=PD^kP^{-1}. Powers of a diagonal matrix are obtained by powering each diagonal entry. diag(32,-1). Chapter 11 · Matrix Powers and Markov Chains
- How do I check inner products?
- An inner product takes two vectors to a scalar and satisfies conjugate symmetry, linearity in the chosen argument convention, and positive definiteness. In real coordinate spaces the standard example is the dot product. ((4)/2)u=(2,2). Chapter 12 · Inner Products, Projections and Gram–Schmidt
- How do I check inconsistent systems and best fit?
- When Ax=b is inconsistent, least squares chooses x̂ minimising ||b−Ax||. It does not make the equations true; it makes the residual as short as possible. The fitted vector A x̂ lies in Col(A). Q^T. Chapter 13 · Least Squares, Orthogonal Matrices, Symmetry and SVD
- How do I check complex vectors and conjugation?
- Complex vectors use conjugation to make squared length real and nonnegative. The conjugate of a+bi is a−bi, and conjugation reverses products componentwise without changing real scalars. Orthogonal: when entries are real, conjugate transpose becomes transpose. Chapter 14 · Complex Linear Algebra, MATLAB and Exam Synthesis
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.
Your AI Mathematics tutor for MAST10007
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