math.la.t.mat.col_space.pivot

Basis theorem: for an n-dimensional vector space any linearly independent set with n elements is a basis, as is any spanning set with n elements; dimension of the column space of a matrix equals the number of pivot columns of the matrix; dimension of the null space of a matrix equals the number of free variables of the matrix

- Created On
- August 25th, 2017
- 5 years ago
- Views
- 4
- Type
- Video
- Language
- English
- Content Type
- text/html; charset=utf-8

The pivot columns of a matrix form a basis for its column space; nullspace of a matrix equals the nullspace of its reduced row-echelon form.

- Created On
- August 25th, 2017
- 5 years ago
- Views
- 4
- Type
- Video
- Language
- English
- Content Type
- text/html; charset=utf-8

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

- License
- GFDL-1.2
- Submitted At
- September 11th, 2017
- 5 years ago
- Views
- 4
- Type
- Language
- English
- Content Type
- text/html