Selected topics in linear algebra:

My notes from Gilbert strang book on Linear algebra : Click Here

Linear Algebra in MATLAB : Download

Visualizations using Geogebra:

1. Equation of plane spanned by vector u=(-1,2,3), v=(5,3,-1).

External Links :

• Linear algebra for Optimization

• http://linear.ups.edu/html/fcla.html Free Download PDF

• http://wwwf.imperial.ac.uk/metric/metric_public/matrices/

Fundamental Spaces in Linear Algebra

In a nutshell

if A is m x n

Column space (R^m): y =Ax, . Vector spanned by linear combination of A. In other words, A transforms a point from R^n to R^m. It is also called as range of A. Therefor, it is a subspace of R^m. If all the points in R^n is mapped to all the points in R^m then the transform is bijective and subspace is the whole space.

Row space (R^n) : Vector spanned by transpose of A, weight for each column of A' comes from x in R^m. In other words, A' takes point from R^m to R^n. It is a subspace of R^n

Null space (kernel): All points x in R^n such that takes y = Ax = 0. It is a subspace of R^n. If x=0 is the only point, then A is consistent and have an Inverse. If there are more than one independent vector in R^n that maps to 0 in R^m then span of that vectors forms a subspace in R^n.

Left Null space : All points in R^m that takes A'x = 0. It is a subspace of R^m

(left) Null space is orthogonal to (row) column space.

Recall the consequences if m=n and rank (dim)=n.

Quick Review at : https://brilliant.org/wiki/fundamental-subspaces/