A First Course in Linear Algebra · The Learning Library

Summary

This course introduces linear algebra through two linked threads: solving systems of simultaneous linear equations with matrices, and using vectors to describe lines, planes and shapes in space. It suits students meeting the subject for the first time, and it builds every idea from worked arithmetic examples before moving to general rules.

The first two chapters, Gaussian Elimination and Solutions to Simultaneous Equations 1, set up the main tool of the course. Students write a system as an array of coefficients and use three elementary row operations to reach row-echelon form. They then read off the answer by back-substitution. The chapters show how to handle a zero pivot by swapping rows. They also show that every system has exactly one of three outcomes: no solution, a unique solution, or infinitely many solutions. Systems containing a parameter $c$ are studied case by case, as are systems with more or fewer equations than unknowns.

Matrices and Algebraic Vectors, Special Matrices and Matrix Inverses turn the arrays into objects with their own algebra. Students learn to add, scale and multiply matrices, and to write any system as $A\mathbf{x}=\mathbf{h}$ . They see that $AB$ usually differs from $BA$ . Special types follow: transposes, with $(AB)^T=B^TA^T$ , along with symmetric, triangular, diagonal, orthogonal and elementary matrices. Inverses are found by applying row operations to $A$ and $I$ side by side. The chapter also covers the $2\times2$ formula and the order-reversing rule $(AB)^{-1}=B^{-1}A^{-1}$ .

Linear Independence and Rank and Determinants give ways to measure whether a matrix is invertible. Rank is the number of non-zero rows left after elimination. A list of vectors is tested for independence by placing the vectors as columns and reducing. Determinants are computed by cofactor expansion, by the $3\times3$ diagonal shortcut, and by elimination, keeping track of how each row operation changes the value. These ideas meet in an Equivalence Theorem: a square matrix is invertible exactly when its rank is full, its columns are independent, $A\mathbf{x}=\mathbf{0}$ has only the zero solution, and $\det A\neq0$ . The adjoint formula $A^{-1}=\frac{1}{\det A}\,\mathrm{adj}\,A$ is also included.

Solutions to Simultaneous Equations 2 revisits systems using the new language. A system is consistent when $A$ and the augmented matrix $[A|\mathbf{h}]$ have the same rank. A consistent square system of rank $r$ needs $p-r$ free parameters in its solution.

The last three chapters, Vectors in Geometry, Straight Lines and Planes and Cross Product, turn to geometry. Topics include position vectors, the Triangle and Parallelogram Laws, the Section Formula and the centroid of a triangle. Lines are written as $\mathbf{r}=\mathbf{a}+t\mathbf{u}$ . The dot product is used to find angles and to give plane equations $ax+by+cz+d=0$ through a normal vector. Students also find where lines and planes intersect, and the distance from a point to a plane. The cross product supplies normal vectors and areas. The scalar triple product gives volumes and a test for whether three vectors are coplanar.

Key ideas:

Gaussian elimination with elementary row operations solves any linear system and shows which of the three outcomes occurs.
Matrix multiplication follows size rules and is not commutative.
An inverse can be found by row-reducing $A$ alongside $I$ , and $(AB)^{-1}=B^{-1}A^{-1}$ .
Rank, linear independence, invertibility and a non-zero determinant are equivalent conditions for a square matrix.
$A\mathbf{x}=\mathbf{h}$ is consistent exactly when $\text{rank}\,A=\text{rank}\,[A|\mathbf{h}]$ .
The dot product gives angles through $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta$ and tests whether two vectors are perpendicular.
Lines and planes have vector and parametric equations, and the points they share are found by solving linear systems.
The cross product is perpendicular to both vectors, and its length $|\mathbf{a}||\mathbf{b}|\sin\theta$ gives areas, while $|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$ gives volumes.

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