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Linear functions

Multiplying a column vector $ X$ by a matrix $ A$ gives a new column vector $ AX$ (possibly of a different dimension, depending on the dimensions of $ A$). If $ A$ is an $ m \times n$ matrix, then $ X$ must be an $ n \times 1$ matrix, or an $ n$-dimensional column vector, and the product $ AX$ will be an $ m
\times 1$ matrix, or an $ m$-dimensional column vector.

You can think of this multiplication as defining a function $ L_A$ from $ n$-dimensional column vectors to $ m$-dimensional column vectors, given by

$\displaystyle L_A(X) = AX.$

Functions of this kind are called linear, and play an important role in mathematics. The main idea behind differentiation is to produce linear approximations (like the tangent line approximation) to functions.

The reason that these functions are called linear is that they satisfy the same two important linearity properties that linear differential operators satisfy: They preserve addition and multiplication by constants.

$\displaystyle L_A(X+Y) = A(X+Y) = AX+AY = L_A(X)+L_A(Y)$

$\displaystyle L_A(cX) = A(cX) = c(AX) = c(L_A(X))$

Example: Suppose

$\displaystyle A = \left(\begin{matrix}1 & 2 \cr 0 & -1 \cr\end{matrix}\right) \...
...end{matrix}\right) \qquad
Y = \left(\begin{matrix}3 \cr -2
\end{matrix}\right).$

Then

$\displaystyle A(X+Y) = A\left( \left(\begin{matrix}1 \cr 2 \end{matrix}\right) ...
...r 0
\cr\end{matrix}\right) =
\left(\begin{matrix}4\cr0 \cr \end{matrix}\right)$

$\displaystyle AX +
AY =
\left(\begin{matrix}1 & 2 \cr 0 & -1 \cr\end{matrix}\ri...
...r2 \cr \end{matrix}\right) =
\left(\begin{matrix}4\cr 0\cr \end{matrix}\right).$

Similarly,

$\displaystyle A(4X) = A\left(4 \left(\begin{matrix}1 \cr 2 \end{matrix}\right) ...
...8
\cr\end{matrix}\right) =
\left(\begin{matrix}20\cr-8 \cr \end{matrix}\right)$

$\displaystyle 4(AX) = 4\left(A\left(\begin{matrix}1 \cr 2 \end{matrix}\right) \...
... \cr
\end{matrix}\right) =
\left(\begin{matrix}20\cr -8\cr \end{matrix}\right).$


next up previous
Next: Homogeneous and non-homogeneous equations Up: Solutions To Matrix Equations Previous: Reading the solutions from
Peter Kostelec
2000-05-05