Lecture Summaries (reverse chronological)

Lecture 23 — Geometric meaning of determinants — Revision
Showed that the absolute value of $\det(A)$, where $A$ is a $2\!\times\! 2$ matrix, is the area of the parallelogram spanned by the two columns of $A$; it is the volume of the parallelepiped in 3D. Sample solution of Question 2(a)-(c) of the MA203 Summer Exam 2015/16, available here.

Lecture 22 — Revision
Sample solutions to Questions 4(a)-(c), 3(a) and 3(c) of the MA203 Summer Exam 2015/16, available here.

Lecture 21 — Properties of determinants
How determinants behave under matrix multiplication (in particular elementary row operations) and multiplication with scalars. A $3\!\times\! 3$ example for finding the eigenvalues using long division of polynomials. The following theorem which sums up the course.

Theroem. For an $n\!\times\! n$ matrix $A$ the following are equivalent.

1. $\det(A)\neq 0$.
2. $A$ is invertible.
3. The columns of $A$ are linearly independent.
4. The columns of $A$ span $\mathbb{R}^n$.
5. The columns of $A$ are a basis of $\mathbb{R}^n$.
6. The (reduced) row echelon form of $A$ has a pivot in every column.
7. The equation $Ax=b$ has a unique solution for every $b\in\mathbb{R}^n$.
8. The null space of $A$ is the zero subspace: $\{(0,0,\ldots,0)\}$.

Lecture 20 — Finding eigenvalues — determinants
Eigenvalues are the solutions of $\det(A-\lambda I)=0$. How to calculate the determinant of a square matrix: development along a row or column, physicists method for $3\!\times\! 3$ matrices.

Lecture 19 — Markov process — steady state
Definitions of a Markov process, state vector, transition matrix and steady state, along with an example.

Lecture 18 — Matrix representation of a linear transformation — cofactor method
Given spanning vectors $b_1,b_2$ and $b_3$ of $\mathbb{R}^3$ and their images under a linear transformation $f$, how one can find the matrix $A$ representing $f$ with respect to the standard basis. The cofactor method for inverting a square matrix.

Lecture 17 — An `exotic' vector space and eigenvectors
The vector space $\mathcal{P}_n$ of polynomials in the variable $x$ of degree at most $n$ with real coefficients, its standard basis and the derivative (with respect to $x$) as a linear transformation from $\mathcal{P}_n$ to $\mathcal{P}_{n-1}$.
Definition of an eigenvector and corresponding eigenvalue of a (square!) matrix $A$, relationship with solutions of $(A-\alpha I)v = 0$.

Lecture 16 — Linear transformations
Definition of a linear transformation and its representation by a matrix with respect to the standard bases of the source and target spaces.

Lecture 15 — Subspaces and bases
Definitions and examples of (i) subspaces of $\mathbb{R}^n$, (ii) the row space, the column space and the null space of a matrix, and (iii) a basis of a subspace. Proposition: $\mathrm{span}\{v_1,\ldots,v_k\}$ has as basis those $v_j$ such that the $j^\mathrm{th}$ column of the row echelon form of the matrix with columns $v_1,\ldots,v_k$ has a pivot.

Lecture 14 — Test for linear independence
Justification of and examples illustrating the fact that a set $\{v_1,\ldots ,v_k\}$ of (column) vectors is linearly independent if and only if the matrix $A$ whose columns are $v_1,\ldots ,v_k$ has a pivot in every column when transformed into row echelon form.

Lecture 13 — Linear span and linear independence
Definitions of $\mathrm{span}\{v_1,\ldots ,v_k\}$, linear combination ($\alpha_1v_1+\cdots+\alpha_kv_k$, $\alpha_i\in\mathbb{R}$) and linear independence of a set $\{v_1,\ldots,v_k\}$ of vectors in $\mathbb{R}^n$. First connections with systems of equations. Special cases: (i) sets of just two vectors and (ii) sets which include the zero vector.

Lecture 12 — More on matrix inversion
Diagonal block matrices and invertibility. One pretty tedious $4\!\times\! 4$ example.

Lecture 11 — Invertibility of certain matrices
Solving matrix equations using inverses. Definition and inversion of diagonal matrices and lower/upper (uni-)triangular matrices.

Lecture 10 — Elementary matrices
Definition of elementary matrices, their connection to elementary row operations and their invertibility. Definition of an invertible matrix and the inverse of a general $2\!\times\! 2$ matrix using row operations.

Lecture 9 — Reduced row echelon form
Definition of reduced row echelon form, pivots, free variables, pivot variables, how to read off solution(s), if any. Criteria for no, a unique or infinitely many solutions.

Lecture 8 — Matrices
Definitions of zero matrix, identity matrix, square matrix, scalar multiple of a matrix. Representation of the complex numbers as $2\!\times\! 2$-matrices of the form $\begin{pmatrix}a & -b\\b & a\end{pmatrix}$. Connection between one linear matrix equation and system of linear equations.

Lecture 7 — Matrices
Definitions, examples and basic properties of products and sums of matrices.

Lecture 6 — Linear Riddles
Solving these linear riddles. Example of system without solution. Recall dot product.

Lecture 5 — Row echelon form
Solving systems of linear equations using row operations (Gaussian elimination) to obtain row echelon form and then back substitution, in particular solutions depending on parameters when there are infinitely many solutions.

Lecture 4 — Systems of linear equations
Definition of a system of linear equations; geometric solution (2 equations in 2 unknowns); solving by manipulating equations; translation into array form and solving by row operations: (i) multiply a row by a non-zero real number and (ii) replace a row by the sum of itself and another row.

Lecture 3 — Parametric descriptions
Parametric descriptions of lines and planes in $\mathbb{R}^3$ and using simultaneous equations to to find points of intersection, if there are any.

Lecture 2 — The dot (or scalar) product
The dot product and its relation to angles between vectors and the norm of a vector. List of properties of the norm and the dot product.
Cauchy‒Schwarz Inequality: $|v\cdot w| \leq \| v\rVert\,\lVert w\rVert$ for $v,w\in\mathbb{R}^n$.

Lecture 1 — Introduction to Vectors
Definitions and geometric interpretations of vector addition, scalar multiple of a vector and the norm of a vector.
Expressions for lines.