Topics in Linear Algebra and Multivariable Calculus
I. Linear spaces (ch. 1):
Review of vector algebra , Vector spaces and subspaces, dependent and linearly
independent sets, bases and dimension, inner products and Euclidean spaces, or-
thogonality and Gram-Schmidt process, projections and best approximation.
II. Linear transformations and Matrices (ch. 2):
Linear transformations, null space and range, algebraic operations on linear
transformations, inverses, matrix representations, systems of linear equations.
III. Diﬀerential calculus (ch. 8/9):
, scalar and vector ﬁelds, open balls and sets, limits and
continuity, derivatives, directional and partial derivatives, higher order derivatives,
chain rule. A ﬁrst order partial diﬀerential equation with constant coeﬃcients,
IV. Determinants (ch. 3):
Volume and orientation, a set of axioms for the determinant function, computa-
tion, existence and uniqueness theorem, product formula, determinants and linear
independence, cofactor expansions and Cramer’s rule.
V. Eigenvalues and Eigenvectors (ch.4):