Unit 1 of 11
Vectors and Matrices
- 5 lessons
- Videos 1 to 5
Vectors and matrices are the basic objects of linear algebra. This unit adds and scales vectors, multiplies a matrix by a vector, and defines linear transformations and their composition.
A vector can be a displacement arrow or a list of measurements, and the same operations apply to both. Linear combinations and span describe which vectors a few chosen directions can reach. A matrix collects columns, and its product with a vector is a weighted sum of those columns. Linearity is then defined by two conditions, and composing two linear maps leads to matrix multiplication, where the order matters.
Every lesson is one video. Each lesson page has the definitions and an example for every key term.
Vectors and Linear Combinations
Vectors and Vector Operations
What a vector is, how a point differs from a displacement, and how to add, subtract and scale vectors as columns, as arrows placed tip to tail and as lists of data, with the zero vector.
Linear Combinations and Span
Linear combinations and their coefficients, the span of a list of generating vectors, how to decide whether a vector lies in a span, and why a redundant generator adds nothing new to the span.
Video 2
Matrices and Linear Transformations
Matrices and Matrix-Vector Multiplication
What an m by n matrix is, how to read its size and entries, how to compute a matrix-vector product as a weighted sum of columns or row by row, and how matrix addition, scalar multiples and the transpose work.
Video 3
Linear Transformations
What makes a map between coordinate spaces linear, how additivity, homogeneity and zero preservation test a rule, why every linear map has a standard matrix, and why nonzero translations fail.
Video 4
Matrix Multiplication and Composition
How composing two linear maps leads to the matrix product, when sizes are compatible, why order matters, and the rules for the identity matrix, associativity, the transpose of a product and blocks.
Video 5