Matrices as linear maps

A matrix is a machine

An $m\times n$ matrix $A$ turns every input vector $\mathbf{x}$ with $n$ entries into an output vector $A\mathbf{x}$ with $m$ entries. The map is linear: $A(a\mathbf{x}+b\mathbf{y})=aA\mathbf{x}+bA\mathbf{y}$, so scaling and adding inputs scales and adds outputs. In finance, if column $j$ of $A$ holds how each of $m$ assets responds to risk factor $j$, then $A\mathbf{x}$ is the vector of asset returns for the factor moves $\mathbf{x}$.

Read the product by columns: $A\mathbf{x}$ is a combination of the columns $\mathbf{a}_j$ of $A$, weighted by the entries of $\mathbf{x}$. Taking $\mathbf{x}=\mathbf{e}_1=(1,0,\dots,0)$ picks out the first column, so the columns of a matrix are the images of the unit vectors. That is all you need to write down any linear map: decide where each unit vector goes and put the results in the columns.

• Linear map: $f(a\mathbf{x}+b\mathbf{y})=af(\mathbf{x})+bf(\mathbf{y})$. Every linear map between finite-dimensional spaces is a matrix.
• Columns: the images of the unit vectors, so $A\mathbf{e}_j$ is column $j$.
• Zero goes to zero: a linear map always sends $\mathbf{0}$ to $\mathbf{0}$.

Lab · what's graded

• From images to a matrix20 pts
• A matrix times a vector15 pts
• Spot the map that is not linear15 pts
• A rotation20 pts
• Matrices and their effects15 pts
• Why columns are images15 pts