University linear algebra: what first-year modules cover and how to practise

What first-year linear algebra modules cover

Applied courses, like MIT's, lean towards computation and applications such as least squares; pure courses, like Oxford's, include proofs such as why every basis of a space has the same size. Your module handbook says which style your exam takes.

The core topics

• Systems of linear equations: elementary row operations, echelon and reduced row echelon form, and when a system has no, one or infinitely many solutions.
• Matrix algebra: multiplication, transpose, inverses, and orthogonal matrices.
• Determinants: properties, computing by row operations, and the link to invertibility and volume.
• Vector spaces: subspaces, span, linear independence, bases, dimension, rank, and the row and column spaces.
• Linear transformations: matrices of a transformation, kernel and image.
• Eigenvalues and eigenvectors: the characteristic polynomial and diagonalisation.
• Orthogonality: dot products, projections, Gram-Schmidt, least squares, and symmetric matrices.

Worked examples

Solving a system by row reduction

Solve x + y + z = 4, x − y + 2z = 9, 2x + y − z = 0. Write the augmented matrix, then use row operations: subtract the first row from the second, and twice the first row from the third, to clear x; then clear y and z in turn. The left-hand side becomes the identity matrix, and the right-hand column reads off the solution: x = 2, y = −1 and z = 3. Check in the first equation: 2 − 1 + 3 = 4.

Determinant and inverse of a 2 × 2 matrix

For A with rows (4, 7) and (2, 6), det A = 4 × 6 − 7 × 2 = 10. Because det A is not zero, A is invertible: swap the diagonal entries, change the signs of the other two and divide by 10, giving A⁻¹ with rows (0.6, −0.7) and (−0.2, 0.4).

Eigenvalues and eigenvectors

For the matrix with rows (4, 1) and (1, 4), the characteristic equation is (4 − λ)² − 1 = 0, so λ² − 8λ + 15 = 0 and the eigenvalues are λ = 3 and λ = 5. The eigenvectors are multiples of (1, −1) and (1, 1), which are orthogonal, as the spectral theorem promises for a symmetric matrix.

Orthogonality

The vectors (2, −3, 1) and (4, 3, 1) are orthogonal because their dot product is 8 − 9 + 1 = 0.

Try a quick matrix-flavoured warm-up with simultaneous equations in our Algebra course, then press Mark to check your answer in the course:

Use the daily paper as practice

Uni-Sine publishes a free daily University Linear Algebra paper. It is a short timed practice paper: 8 questions worth 25 to 35 marks, with a suggested 40 to 55 minutes, marked when you finish. It is not your university's exam, so use it alongside your problem sheets and past papers.

• What it covers well: matrix operations, inverses, determinants, systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, eigenvalues and eigenvectors, and orthogonality.
• What it does not cover: the proofs that pure courses expect, and later topics such as the spectral theorem, quadratic forms and positive definite matrices in depth.

Common mistakes in linear algebra

• Multiplying matrices in the wrong order. AB and BA are usually different.
• Dividing by a matrix. Multiply by the inverse instead, and only if it exists.
• Calling the zero vector an eigenvector. Eigenvectors must be non-zero.
• Confusing span and independence. A spanning set can contain redundant vectors; a basis cannot.

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