Tartalomjegyzék

Fundamental theorem of LA

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File: d2017_10_05_gyak4_basicthm_LA.mlx

Author: Peter Polcz ppolcz@gmail.com

Created on 2017. October 18.

A = rand(3,2);
A = A * A'

Output:

A =
    1.5932    0.7144    0.8144
    0.7144    0.7547    0.0769
    0.8144    0.0769    0.6076

Matrix $A$ is rank deficient:

rank(A)

Output:

ans =
     2

Image space or range of matrix $A$ : $\mathrm{Im}(A)$

orth(A)

Output:

ans =
   -0.8249    0.0850
   -0.3928   -0.7971
   -0.4065    0.5978

Null space or kernel of matrix $A^T$: $\mathrm{Ker} \left(A^T\right)$

null(A')

Output:

ans =
   -0.5588
    0.4586
    0.6909

Due to the fundamental theorem of linear algebra, $\mathrm{Im}(A) \perp \mathrm{Ker} \left(A^T\right)$

Visual demonstration

figure, hold on, view(3)
pcz_plotvec(orth(A))
pcz_plotvec(null(A'))

Published with MATLAB® R2017b, and sanitarized by BeautifulSoup (Python script using BSoup is written by Polcz).