Transposes, Permutation, Vector Spaces

May 23, 2019

PA=LU

To extend and finish the factorization of $A=LU$. As matrix $A$ may not be in the ideal form of all pivots are non-zero, we need to do $PA$ to exchange rows into their pivot positions.

Transposes and Symmetry

We know that transposing a matrix is switching its columns with rows, that is, $A_{ij}^T$ is $A_{ji}$. A symmetric matrix is first a square matrix and $S_{ij}=S_{ji}, \forall i,j$ or equivalently $S^\top =S$. For example,

is a symmetric matrix.

Prop 1. For any given rectangular matrix $R$, $R^TR$ is always symmetric.

Let $R=\left[\begin{matrix} 1 & 2 & 4\\ 3 & 3 & 1 \end{matrix}\right]$, then

We see for this particular $R$, $(R^TR)^T=R^TR$. More formally, we can use the lemma that $(AB)^T=B^TA^T$ from lecture 4 and $(R^TR)^T=R^T(R^T)^T=R^TR$ and we can conclude $R^TR$ is always symmetric. Same for $RR^T$.

Vector Spaces

While we do not list the formal definition of vector spaces and the requirement a space should follow here, an intuitive definition is that a vector space is any space we can perform linear combinations on vectors. Linear combinations are we can add vectors, multiply a vector with a constant. Also remember we also need to be able to multiply a vector with 0.

The most common vector space is $\mathbb{R}^2$, the set of (column) vectors with two real numbers. Examples are $\left[\begin{matrix} 2\\ -1 \end{matrix}\right]\left[\begin{matrix} 0\\ 0 \end{matrix}\right]\left[\begin{matrix} \pi \\ e \end{matrix}\right]$. Another exmaple is $\mathbb{R}^n$.

Subspaces

A vector space that is contained inside another vector space is a $subspace$ of that space. Any lines pass through origin (Note vectors start from origin) in $\mathbb{R}^2$ is a subspace of $\mathbb{R}^2$. Lines that do not pass through origin are not because when you multiply a vector on this line with $0$ it becomes a zero vector, and this is not on that line.

The subspaces of $\mathbb R^2$ are:

1. all of $\mathbb R^2$
2. any line through origin
3. the zero vector itself($Z$)

The subspaces of $\mathbb R^3$ are:

1. all of $\mathbb R^3$
2. any planes through origin
3. any lines through origin
4. zero vector

Column Space

The space formed by the columns of a matrix is the column space of this vector. Let

the column space of $A$, $C(A)$ is the plane containing vector $\left[\begin{matrix} 1\\ 2\\ 4 \end{matrix}\right]$ and $\left[\begin{matrix} 3\\ 3\\ 1 \end{matrix}\right]$, and undoubtedly this plane goes through origin.