Notes on Strang Lecture - 5

Permutation matrices and transposes

Permutation matrices

A permutation matrix $PA$ executes row exchanges of $A$.

For LU decomposition with row exchanges, we have:

Key property: For any permutation matrix $P$:

The matrix is orthogonal.

Matrix transpose

If $A = [a_{ij}]$, then $A^T = [a_{ji}]$.

Symmetric matrices

A matrix $A$ is symmetric if $A = A^T$

Vector subspaces

Definition

A vector subspace is a subset of the vectors in a vector space, that is itself a vector space under the same operations.

Examples in $\mathbb{R}^2$ and $\mathbb{R}^3$

All possible subspaces of $\mathbb{R}^2$:

1. The zero vector $\{\vec{0}\}$ (dimension 0)
2. Any line through the origin (dimension 1)
3. The entire space $\mathbb{R}^2$ itself (dimension 2)

All possible subspaces of $\mathbb{R}^3$:

1. The zero vector $\{\vec{0}\}$ (dimension 0)
2. Any line through the origin (dimension 1)
3. Any plane through the origin (dimension 2)
4. The entire space $\mathbb{R}^3$ itself (dimension 3)

Important note: A line or plane that does not pass through the origin is not a subspace, as it fails to contain the zero vector and is not closed under scalar multiplication.