Linear Transformations (applied to, and producing 2D vectors)

$F$ is a linear transformation (operating on vectors) if and only if, for all scalars $c$ and vectors $\bar{x}$ and $\bar{y}$ we have:

  1. $F(c\bar{x}) = cF(\bar{x})$
  2. $F(\bar{x}+\bar{y}) = F(\bar{x}) + F(\bar{y})$

An amazing consequence of this definition is that knowledge of what a specific linear transformation (operating on two-dimensional vectors) does to the vectors $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$ is sufficient to determine what that transformation does to any (two-dimensional) vector. As an example, suppose we know that for a given linear transformation, $M$, we have
$$M\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}4\\5\end{pmatrix}$$
$$M\begin{pmatrix}0\\1\end{pmatrix} = \begin{pmatrix}-3\\7\end{pmatrix}$$
Now suppose we wish to know what $M$ does to an arbitrary vector, say $\begin{pmatrix}-6\\8\end{pmatrix}$:
$$\begin{align*}
M\begin{pmatrix}-6\\8\end{pmatrix} &= M \left( \begin{pmatrix}-6\\0\end{pmatrix} + \begin{pmatrix}0\\8\end{pmatrix} \right) \\
&= M\begin{pmatrix}-6\\0\end{pmatrix} + M\begin{pmatrix}0\\8\end{pmatrix} \quad \textrm{...by property (2)}\\
&= M\left( -6 \begin{pmatrix}1\\0\end{pmatrix} \right) + M \left( 8 \begin{pmatrix}0\\1\end{pmatrix} \right) \\
&= -6 M\begin{pmatrix}1\\0\end{pmatrix} + 8 M \begin{pmatrix}0\\1\end{pmatrix} \quad \textrm{...by property (1)}\\
&= -6 \begin{pmatrix}4\\5\end{pmatrix} + 8 \begin{pmatrix}-3\\7\end{pmatrix} \\
&= \begin{pmatrix}(-6) \cdot 4\\ (-6) \cdot 5\end{pmatrix} + \begin{pmatrix} 8 \cdot (-3)\\8 \cdot 7\end{pmatrix} \\
&= \begin{pmatrix}(-6) \cdot 4 + 8 \cdot (-3)\\ (-6) \cdot 5 + 8 \cdot 7\end{pmatrix} \\
&= \begin{pmatrix}-48\\26\end{pmatrix}
\end{align*}$$

Because we can completely characterize a linear transformation in this way -- through knowledge of where it sends $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$ -- we "name" a linear transformation (which is both applied to, and produces two-dimensional vectors) with a matrix whose columns represent the output vectors corresponding to inputs $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$, respectively.

Notice below, how much more efficiently we can perform the same evaluation found a moment ago, by using the matrix form:
$$M\begin{pmatrix}-6\\8\end{pmatrix} = \begin{bmatrix}4 & -3\\5 & 7\end{bmatrix} \begin{pmatrix}-6\\8\end{pmatrix} = \begin{pmatrix}4 \cdot (-6) + (-3) \cdot 8\\5 \cdot (-6) + 7 \cdot 8\end{pmatrix} = \begin{pmatrix}-48\\26\end{pmatrix}$$
In general, if we have a linear transformation, $M=\begin{bmatrix}a & b\\c & d\end{bmatrix}$,
$$M\begin{pmatrix}x\\y\end{pmatrix} = \begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}ax+by \\ cx + dy\end{pmatrix}$$

When are Combinations of Linear Transformations also Linear Transformations?

Suppose $F$ and $G$ are linear transformations.

Is the sum, $F+G$, a linear transformation? To answer this question, let us check the two defining properties of a linear transformation.

  1. Is $(F+G)(c \bar{x}) = c(F+G)(\bar{x})$ ?

    Yes! Consider the following:
    $$\begin{align*}
    (F+G)(c \bar{x}) &= F(c \bar{x}) + G(c \bar{x})\\
    &= cF(\bar{x}) + cG(\bar{x})\\
    &= c(F(\bar{x}) + G(\bar{x}))\\
    &= c(F+G)(\bar{x})
    \end{align*}$$

  2. Is $(F+G)(\bar{x} + \bar{y}) = (F+G)(\bar{x}) + (F+G)(\bar{y})$ ?

    Yes! Consider the following:
    $$\begin{align*}
    (F+G)(\bar{x}+\bar{y}) &= F(\bar{x}+\bar{y})+G(\bar{x}+\bar{y})\\
    &= F(\bar{x}) + F(\bar{y}) + (G(\bar{x}) + G(\bar{y}))\\
    &= F(\bar{x}) + G(\bar{x}) + F(\bar{y}) + G(\bar{y})\\
    &= (F+G)(\bar{x}) + (F+G)(\bar{y})
    \end{align*}$$

So the sum of two linear transformations is itself a linear transformation!


What about the difference, $F-G$? Is this a linear transformation? Let us again check the two defining properties of a linear transformation:

  1. Is $(F-G)(c \bar{x}) = c(F-G)(\bar{x})$ ?

    Yes! Consider the following:
    $$\begin{align*}
    (F-G)(c \bar{x}) &= F(c \bar{x}) - G(c \bar{x})\\
    &= cF(\bar{x}) - cG(\bar{x})\\
    &= c(F(\bar{x}) - G(\bar{x}))\\
    &= c(F-G)(\bar{x})
    \end{align*}$$

  2. Is $(F-G)(\bar{x} + \bar{y}) = (F-G)(\bar{x}) + (F-G)(\bar{y})$ ?

    Yes! Consider the following:
    $$\begin{align*}
    (F-G)(\bar{x}+\bar{y}) &= F(\bar{x}+\bar{y})-G(\bar{x}+\bar{y})\\
    &= F(\bar{x}) + F(\bar{y}) - (G(\bar{x}) + G(\bar{y}))\\
    &= F(\bar{x}) - G(\bar{x}) + F(\bar{y}) - G(\bar{y})\\
    &= (F-G)(\bar{x}) + (F-G)(\bar{y})
    \end{align*}$$

So the difference of two linear transformations is itself a linear transformation!


What about the product, $FG$? Is this a linear transformation? Let us again check the two defining properties of a linear transformation:

  1. Is $(FG)(c \bar{x}) = c(FG)(\bar{x})$ ?

    Consider the following:
    $$\begin{align*}
    (FG)(c \bar{x}) &= F(c \bar{x}) \cdot G(c \bar{x})\\
    &= cF(\bar{x}) \cdot cG(\bar{x})\\
    &= c^2 (F(\bar{x}) \cdot G(\bar{x}))\\
    &= c^2 (FG)(\bar{x})\\
    &\neq c (FG)(\bar{x}) \textrm{ for all values of $c$}
    \end{align*}$$

So the product of two linear transformations is not a linear transformation!


What about the composition, $F \circ G$? Is this a linear transformation? Let us again check the two defining properties of a linear transformation:

  1. Is $(F \circ G)(c \bar{x}) = c(F \circ G)(\bar{x})$ ?

    Yes! Consider the following:
    $$\begin{align*}
    (F \circ G)(c \bar{x}) &= F(G(c\bar{x}))\\
    &= F(cG(\bar{x}))\\
    &= cF(G(\bar{x}))\\
    &= c(F \circ G)(\bar{x})
    \end{align*}$$

  2. Is $(F \circ G)(\bar{x} + \bar{y}) = (F \circ G)(\bar{x}) + (F \circ G)(\bar{y})$ ?

    Yes! Consider the following:
    $$\begin{align*}
    (F \circ G)(\bar{x} + \bar{y}) &= F(G(\bar{x} + \bar{y}))\\
    &= F(G(\bar{x}) + G(\bar{y}))\\
    &= F(G(\bar{x})) + F(G(\bar{y}))\\
    &= (F \circ G)(\bar{x}) + (F \circ G)(\bar{y})
    \end{align*}$$

So the composition of two linear transformations is itself a linear transformation!


So the sum, difference, and composition of two linear transformations are themselves linear transformations. Consequently, if we are talking about linear transformations operating on two-dimensional vectors, then we can also say that the sum, difference, and composition of two linear transformations can be written as a matrix, whose first and second columns are determined by where the vectors $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$ are taken under each, respectively.

This gives us a way to find the matrix form for the sum, difference, and composition of two linear transformations (operating on two-dimensional vectors) directly from the matrix forms for the linear transformations being combined.


Consider the sum of two $2 \times 2$ matrices:

$$\begin{align*}
\left( \begin{bmatrix}a & b\\c & d\end{bmatrix}+ \begin{bmatrix}e & f\\g & h\end{bmatrix} \right) \begin{pmatrix}1\\0\end{pmatrix} &= \begin{bmatrix}a & b\\c & d\end{bmatrix}\begin{pmatrix}1\\0\end{pmatrix} + \begin{bmatrix}e & f\\g & h\end{bmatrix}\begin{pmatrix}1\\0\end{pmatrix} \\
&= \begin{pmatrix}a\\c\end{pmatrix} + \begin{pmatrix}e\\g\end{pmatrix} \\
&= \begin{pmatrix}a+e\\c+g\end{pmatrix}
\end{align*}$$
while,
$$\begin{align*}
\left( \begin{bmatrix}a & b\\c & d\end{bmatrix}+ \begin{bmatrix}e & f\\g & h\end{bmatrix} \right) \begin{pmatrix}0\\1\end{pmatrix} &= \begin{bmatrix}a & b\\c & d\end{bmatrix}\begin{pmatrix}0\\1\end{pmatrix} + \begin{bmatrix}e & f\\g & h\end{bmatrix}\begin{pmatrix}0\\1\end{pmatrix} \\
&= \begin{pmatrix}b\\d\end{pmatrix} + \begin{pmatrix}f\\h\end{pmatrix} \\
&= \begin{pmatrix}b+f\\d+h\end{pmatrix}
\end{align*}$$
Thus, we can describe the sum of the two linear transformations (operating on two-dimensional vectors) in matrix form as:
$$\begin{bmatrix}a & b\\c & d\end{bmatrix}+ \begin{bmatrix}e & f\\g & h\end{bmatrix} = \begin{bmatrix}a+e & b+f\\c+g & d+h\end{bmatrix}$$


The difference of two linear transformations (operating on two-dimensional vectors) can be found in an almost identical way, yielding:
$$\begin{bmatrix}a & b\\c & d\end{bmatrix}- \begin{bmatrix}e & f\\g & h\end{bmatrix} = \begin{bmatrix}a-e & b-f\\c-g & d-h\end{bmatrix}$$


The composition of two such transformations proves a bit more interesting...

$$\begin{align*}
\left( \begin{bmatrix}a & b\\c & d\end{bmatrix} \circ \begin{bmatrix}e & f\\g & h\end{bmatrix} \right) \begin{pmatrix}1\\0\end{pmatrix} &= \begin{bmatrix}a & b\\c & d\end{bmatrix} \left( \begin{bmatrix}e & f\\g & h\end{bmatrix} \begin{pmatrix}1\\0\end{pmatrix} \right) \\
&= \begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{pmatrix}e\\g\end{pmatrix}\\
&= \begin{pmatrix}ae + bg\\ce + dg\end{pmatrix}
\end{align*}$$
while,
$$\begin{align*}
\left( \begin{bmatrix}a & b\\c & d\end{bmatrix} \circ \begin{bmatrix}e & f\\g & h\end{bmatrix} \right) \begin{pmatrix}0\\1\end{pmatrix} &= \begin{bmatrix}a & b\\c & d\end{bmatrix} \left( \begin{bmatrix}e & f\\g & h\end{bmatrix} \begin{pmatrix}0\\1\end{pmatrix} \right) \\
&= \begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{pmatrix}f\\h\end{pmatrix}\\
&= \begin{pmatrix}af+bh\\cf+dh\end{pmatrix}
\end{align*}$$
Thus, we can describe the composition of the two linear transformations (operating on two-dimensional vectors) in matrix form as:
$$\begin{bmatrix}a & b\\c & d\end{bmatrix}\circ \begin{bmatrix}e & f\\g & h\end{bmatrix} = \begin{bmatrix}ae+bg & af+bh\\ce+dg & cf+dh\end{bmatrix}$$
Notice, somewhat unexpectedly -- the method of combining matrices above is typically described as matrix "multiplication" even though a "composition" lies at its heart!