0.2 Matrices

0.2.1 Rectangular arrays

A matrix is an $m$-by-$n$ array of scalars from a field $F$. If $m=n$, the matrix is said to be square. The set of all matrices over $F$ is denoted $M_{m,n}(F)$, or just $M_n(F)$.

0.2.2 Linear transformations

Let $U$ be an $n$-dimensional vector space and let $V$ be an $m$-dimensional vector space, both over the same field of $F$. let ${\beta }_U$ be a basis of $U$ and ${\beta }_V$ be a basis of $V$.

A linear transformation is a function $T$ : $U\to V$ such that $T(a_1{x}_1+a_2{x}_2)=a_{1}T(x_1)+a_{2}T(x_2)$ for any scalars $a_1,a_2$ and any vectors $x_1,x_2$.

A matrix $A$ can represent a linear transformation $T$, but it will depend on the bases chosen.

0.2.3 Vector spaces associated with a matrix or linear transformation

Any $n$-dimensional vector space over $F$ may be identified with $F^n$. we may think of $A\in M_{m,n}(F)$ as a linear transformation $x\to Ax$ from $F^n$ to $F^m$. The domain of the linear transform is $F^n$ and its range is $F^m$. its nullspace is $A=x\in F^n:Ax=0$.

The dimension of nullspace $A$ is denoted as nullity $A$ The dimension of range $A$ is denoted as rank $A$

The rank nullity theorem is as follows:

dim(range $A$) + dim(nullspace $A$) = rank $A$ + nullity $A$ = $n$

0.2.4 Matrix operations

Matrix addition is entrywise

Matrix multiplication is only defined when the internal dimensions align. i.e. $A(m,n)\ast B(n,q)$.

0.2.5 The transpose, conjugate, transpose, and trace

The transpose is denoted as $A^T$.

The conjugate is denoted by $A^{\ast }$ and is the transpose of an entrywise conjugate.

0.2.7 Column space and row space of a matrix

The range of $A$ is also called its column space because $Ax$ is a linear combination of the columns of $A$ for any $x\in F^n$. The row space of $A$ is the same, except for $y^T\in F^m$.

If the column space of $A_{m,n}$ is contained in the column space of $B_{m,k}$ there is some $X_{k,n}$ such that $A=BX$.

0.2.8 The all-ones matrix and vector