Different perspectives of matrices

Overview §

Matrices are used to visualize linear transformations. In particular, an $m\times n$ matrix corresponds to a linear map from an $n$-dimensional space to an $m$-dimensional space.

There are three perspectives for understanding matrices. The “entries” perspective is a computation of the “row” and “column” perspectives.

The column perspective emphasizes that the columns of matrix $M$ tell you where $A_M:F^n\to {\mathbb{F}}^m$ sends a standard basis vector $e_i\in {\mathbb{F}}^n$. Each column of a $M$ is a vector $v_j=A_M{e}_i\in F^m$.

$$M=\left(\begin{array}{c}\mid \cdots \mid \\ v_1\cdots v_n\\ \mid \cdots \mid \end{array}\right)$$

The row perspective emphasizes that the row of matrix $M$ give individual entries of the output $p_i{A}_M:{\mathbb{F}}^n\to \mathbb{F}$. Each row is a linear function $f_1,\cdots ,f_m:{\mathbb{F}}^n\to \mathbb{F}$, and the $i$th component of $A_M(v)$ is $f_i(v)=p_i{A}_M(v)$.

$$M=\left(\begin{array}{c}\text{textemdash}{f}_i\text{textemdash}\\ \cdots \cdots \cdots \\ \text{textemdash}{f}_m\text{textemdash}\end{array}\right)$$

A particular entry of $M$ is therefore the $i$th component of a column vector, given by $a_{ij}=p_i(A_M{e}_j)$. Explicitly, $p_i$ is a linear function called the projection to the $i$th coordinate. The entry perspective is useful for matrix multiplication.

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Matrix definition §

Definition: Matrix

An $m\times n$ matrix over the field $\mathbb{F}$ is a two-dimensional array $M$ containing elements of $\mathbb{F}$ which as $m$ rows and $n$ columns.

$$M=\left(\begin{array}{c}{a}_{11}\cdots a_{1n}\\ \cdots \cdots \cdots \\ a_{m1}\cdots a_{mn}\end{array}\right)$$

The The entry in row $i$ and column $j$ of the matrix is denoted $a_{ij}\in \mathbb{F}$.

Two matrices $M,M^{\prime }$ are equal if they are both $m\times n$ and all entries $M_{ij}=M_{ij}^{\prime }$.

In terms of matrices, a linear map is explicitly a transformation $A_M:{\mathbb{F}}^n\to {\mathbb{F}}^m$ associated with some matrix, defined by

$$A_M\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=\left(\begin{array}{c}{a}_{11}{x}_1+\cdots +a_{1n}{x}_n\\ \cdots \\ a_{m1}{x}_1+\cdots +a_{mn}{x}_n\end{array}\right).$$

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The column perspective §

$$M=\left(\begin{array}{c}\mid \cdots \mid \\ v_1\cdots v_n\\ \mid \cdots \mid \end{array}\right)$$

The underlying idea behind the “column perspective” is that linear maps are determined by what they do to a basis. Thus, to understand what $A_M$ does to ${\mathbb{F}}^n$, we compute $A_M{e}_j$ for each vector in the standard basis $(e_1,\cdots ,e_n)$.

• (Lemma 58) If $M$ is an $m\times n$ matrix, then $A_M{e}_j$ is the $j$th column of $M$ so that

$$A_M{e}_j=A_M\left(\begin{array}{c}0\\ \cdots \\ x_j=1\\ \cdots \\ 0\end{array}\right)=\left(\begin{array}{c}{a}_{11}(0)+\cdots +a_{1j}(1)+\cdots +a_{1n}(0)\\ \cdots \\ a_{m1}(0)+\cdots +a_{mj}(1)+\cdots +a_{mn}(0)\end{array}\right)=\left(\begin{array}{c}{a}_{1j}\\ a_{2j}\\ \cdots \\ a_{mj}\end{array}\right).$$

• If $x$ is a general vector, then $A_{M}x$ is a linear combination of the columns of $A$ with “weights” provided by the entries of $x$.

$$A_{M}x=A_M(x_1{e}_1+\cdots +x_n{e}_n)=x_1{A}_M{e}_1+\cdots +x_n{A}_M{e}_n=x_1{v}_1+\cdots +x_n{v}_n$$

• (Corollary 59) If $M$ is an $m\times n$ matrix with columns $v_1,\cdots ,v_n$, then the associated linear map $A_M:{\mathbb{F}}^n\to {\mathbb{F}}^m$ has:
  • $\text{im}(A_M)=$ the span of all columns, $\text{span}(v_1,\cdots ,v_n)$
  • $\text{ker}(A_M)=$ the set of linear relations between the columns, $\text{Rel}(v_1,\cdots ,v_n)$
  • Further, by rank-nullity, $\text{rank}(A_M)=$ the # of non-redundant columns, and $\text{null}(A_M)=$ the # of redundant columns.
• (Corollary 60) If $A_M:{\mathbb{F}}^n\to {\mathbb{F}}^m$ is any linear map, then there exists a unique $m\times n$ matrix $M$ so that $A=A_M$.
  • In other words, matrices and linear maps are a “dictionary”; any linear map can be “worked with” in the language of matrices.

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The row perspective §

$$M=\left(\begin{array}{c}\text{textemdash}{f}_i\text{textemdash}\\ \cdots \cdots \cdots \\ \text{textemdash}{f}_m\text{textemdash}\end{array}\right)$$

A row vector is a $1\times n$ matrix corresponding to a linear function $f:{\mathbb{F}}^n\to \mathbb{F}$. Its entries are number outputs $f(e_j)$ for each $j$.#wip in what?

Given such a row vector, as well as a column vector $\vec{v}\in {\mathbb{F}}^n$, the pairing $f(\vec{v})$ of the two vectors is given by

$$f(\vec{v})=(a_1\cdots a_n)\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=a_1{x}_1+\cdots +a_n{x}_n.$$

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The entry perspective §

Definition: Coordinate projection

The projection to the $i$th coordinate is a linear function $p_1,\cdots ,p_n:{\mathbb{F}}^n\to \mathbb{F}$ defined by

$$p_i\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=(0\cdots a_1=1\cdots 0)\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=x_i.$$

That is, the projection outputs the value of the $i$th coordinate.

• (Lemma 61) If $A_M:{\mathbb{F}}^n\to {\mathbb{F}}^m$ is the linear map associated with an $m\times n$ matrix $M$, then for $1\le i\le m$, the $i$th row of $M$ is the row vector $(a_{i1}\cdots a_{in})$ corresponding to the linear map $p_i{A}_M:{\mathbb{F}}^n\to {\mathbb{F}}^m\to \mathbb{F}$.

$$p_i{A}_M\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=a_{i1}{x}_1+\cdots a_{in}{x}_n$$

• (Lemma 62) If $M=(a_{ij})$ is an $m\times n$ matrix and $A_M:{\mathbb{F}}^n\to {\mathbb{F}}^m$ is an associated linear map, then the particular entry $a_{ij}=p_i(A_M{e}_j)$.
  • The entry $a_{ij}$ is the $i$th component of the column vector $A_M{e}_j$.