 This subject provides an introduction to linear algebra and multivariable calculus. We cover topics in linear algebra including solving systems of linear equations, matrices, LU factorisation, determinants, vector spaces, linear independence, basis, fundamental spaces, projection, eigenvectors, and diagonalisation with a balanced emphasis on proofs and computation. Subsequently, this subject extends the notion of derivation and integration for multivariable functions by introducing the gradient, directional derivatives, implicit functions, maxima and minima, multivariable integration, curvilinear coordinates, and change of variables.

#### Learning Objectives

At the end of the term, students will be able to:

• Express linear models in vector-matrix form
• Solve linear systems by elimination and by inverse matrices
• Estimate a solution of an overdetermined linear system
• Compute the rank and nullity of a matrix
• Use the Fundamental Theorem of Linear Algebra to characterise the solutions of a linear system
• Compute the determinant through cofactor expansion
• Compute the eigenvalues and eigenvectors of a matrix
• Compute partial derivatives and linear approximations to multivariable functions
• Connect functions, gradients, and level curves
• Find critical points of multivariable functions and classify them as min/max/saddle
• Set up and evaluate integrals of multiple variables

5-0-7