Vectors in n dimensions

7 lessons

Vectors with any number of entries, dot products and length, combinations, span, independence, subspaces, basis and dimension.

Review first: Vector addition and scalar multiplication The dot product and the angle between vectors Orthogonal vectors Adding, subtracting and scaling matrices

1.1Adding and scaling vectors in n dimensions

1.2Dot product, length and orthogonality in n dimensions

1.3Linear combination

1.4Span

1.5Linear independence

1.6Subspaces

1.7Basis and dimension

Linear systems and row reduction

9 lessons

Row reduction taught again for any size of system, then reduced echelon form, free variables, Ax as a combination of columns, solution sets and systems in use.

2.1Reduced row echelon form

2.2Rank

2.3Multiplying a matrix by a vector

2.4Homogeneous systems and solutions in parametric vector form

2.5Linear systems in use: networks, chemical equations, fitting curves

2.6Writing a system as a matrix equationretaught

2.7Augmented matrices and row operationsretaught

2.8Echelon forms and Gaussian eliminationretaught

2.9Solving a system by row reduction: one, none or infinitely many solutionsretaught

Matrix algebra and linear transformations

10 lessons

Matrix multiplication taught again, a matrix as a linear map, composing maps, the transpose, inverses by row reduction, kernel and image, rank and nullity, and change of basis.

Review first: Rows, columns and dimensions Inverse of a 2 by 2 matrix, and solving with it

3.1Linear transformation

3.2Finding the matrix of a transformation

3.3Composing transformations as a matrix product

3.4The transpose and symmetric matrices

3.5Finding an inverse by row reducing A next to I

3.6Kernel

3.7Image

3.8Rank and nullity

3.9Coordinates in a basis and change of basis

3.10Matrix multiplicationretaught