linear algebra i
The course starts with basic properties of linear equations and matrices, which felt familiar to what we'd done at school. Then it gets abstract — and the surprise was that the abstraction doesn't replace the matrices, it explains them. A linear map exists on its own; a matrix is what you get once you've picked a basis, and the change of basis theorem says which matrix you got was never the point.
Defining things in this way not only gives us some nice properties, but allows us to notice them popping up in other areas of maths, for example the vector space of sequences or polynomials with degree , that we see in Analysis I.
syllabus
Systems of linear equations. Matrices and the beginnings of matrix algebra. Use of matrices to describe systems of linear equations. Elementary Row Operations (EROs) on matrices. Reduction of matrices to echelon form. Application to the solution of systems of linear equations.
Inverse of a square matrix. Reduced row echelon (RRE) form and the use of EROs to compute inverses; computational efficiency of the method. Transpose of a matrix; orthogonal matrices.
Vector spaces: definition of a vector space over a field (such as , , ). Subspaces. Many explicit examples of vector spaces and subspaces.
Span of a set of vectors. Examples such as row space and column space of a matrix. Linear dependence and independence. Bases of vector spaces; examples. The Steinitz Exchange Lemma; dimension. Application to matrices: row space and column space, row rank and column rank. Coordinates associated with a basis of a vector space.
Use of EROs to find bases of subspaces. Sums and intersections of subspaces; the dimension formula. Direct sums of subspaces.
Linear transformations: definition and examples (including projections associated with direct-sum decompositions). Some algebra of linear transformations; inverses. Kernel and image, Rank-Nullity Theorem. Applications including algebraic characterisation of projections (as idempotent linear transformations).
Matrix of a linear transformation with respect to bases. Change of Bases Theorem. Applications including proof that row rank and column rank of a matrix are equal.
Bilinear forms; real inner product spaces; examples. Mention of complex inner product spaces. Cauchy–Schwarz inequality. Distance and angle. The importance of orthogonal matrices.