Linear Algebra

Matrix

Super foundational knowledge.

Concepts

• Fundamental Subspaces of a Matrix
• Determinant
• Linear Transformation
• Rotation Matrix
• Eigenvalues and Eigenvectors
• Complex Matrix

Elementary Row Operations (EROs)

• Swap two rows $R_1\leftrightarrow R_2$
• Add a scalar multiple of one row to another $R_1+k{R}_3\to R_1$
• Multiply any row by a nonzero scalar $k{R}_2\to R_2$

We say that two systems are equivalent if they have the same solution set.

Row Echelon Form (REF)

• A matrix is in Row Echelon Form (REF) if

1. All rows whose entries are all zero appear below all rows that contain nonzero entries,
2. Each leading entry is to the right of the leading entries above it.

Reduced Row Echelon Form (RREF)

• A matrix is in Reduced Row Echelon Form (RREF) if it is in REF and (3) Each leading entry is a 1 called a leading one, (4) Each leading one is the only nonzero entry in its column.

Rank

The rank of a matrix A, denoted by rank(A), is the number of leading entries in any REF of A (excluding $\vec{b}$).

Dimension

The dimension is the number of elements in a basis. Definition 22.5. If $B=\{\overrightarrow{v_1},\dots ,\overrightarrow{v_k}\}$ is a basis for a subspace $\mathbb{S}$ of ${\mathbb{R}}^n$, then dim($\mathbb{S}$) = $k$. If $\mathbb{S}$ =$\{\vec{0}\}$, then dim($\mathbb{S}$) = 0 since $\varnothing$ is a basis for $\mathbb{S}$.

Matrix Multiplication

IMPORTANT: Matrix multiplication is not commutative.

Matrix Inverse

Definition 30.1. Let $A\in M_{n\times n}(\mathbb{R})$. If there exists a $B\in M_{n\times n}(\mathbb{R})$ such that $AB=I=BA$ then $A$ is invertible and $B$ is an inverse of A (and B is invertible with A an inverse of B).

Unique Inverses

If $B$ and $C$ are both inverses of $A$, then $B=C$.

Properties of matrix inverses. Let $A,B\in M_{n\times n}(\mathbb{R})$ be invertible and let $c\in \mathbb{R}$ with $c\ne 0$. Then $(cA{)}^{-1}=\frac{1}{c}{A}^{-1}$ $(AB{)}^{-1}=B^{-1}{A}^{-1}$ $(A^k{)}^{-1}=(A^{-1}{)}^k\text{for a positive integer k}$ $(A^T{)}^{-1}=(A^{-1}{)}^T$ $(A^{-1}{)}^{-1}=A$

For all $B,C\in M_{n\times k}(\mathbb{R})$, if $AB=AC$, then $B=C$ (left cancellation) For all $B,C\in M_{k\times n}(\mathbb{R})$, if $BA=CA$, then $B=C$ (right cancellation)

Diagonal Matrix / Diagonalization

An $n\times n$ matrix A is diagonalizable if there exists an $n\times n$ invertible matrix $P$ and an $n\times n$ diagonal matrix $D$ so that $P^{-1}AP=D$. In this case, we say that P diagonalizes $A$ to $D$.

The trace of a square matrix is the sum of its diagonal entries. $tr(A)=a_{11}+\cdots +a_{nn}$

An $n\times n$ matrix A is diagonalizable if and only if $a_{\lambda }=g_{\lambda }$ for every eigenvalue of A.

Tridiagonal Matrix

Saw this in CS370.

Powers of Matrices

Let $A$ be an $n\times n$ diagonalizable matrix. Then $P^{-1}AP=D$ for some $n\times n$ invertible matrix $P$ and $n\times n$ diagonal matrix $D$. We have than

$A^k=P{D}^k{P}^{-1}$