Linear transformation formula

Question

How to find formula for linear transformation $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}^4$ when the following is given: $$\varphi ((5,1))=(2,5,1,1)$$ $$\varphi((1,0))=(3,4,2,2)$$ What is the method for solving such questions?

Answer 1

Hints:

$$\phi((x,y)) = \phi((x,0) + (0,y)) = \phi((x,0)) + \phi((0,y)) = x\phi((1,0)) + y\phi((0,1))$$

$$\phi((0,1)) = \phi((-5,0) + (5,1)) = -5\phi((1,0)) + \phi((5,1))$$

Answer 2

This linear transformation would be represented by a $4\times2$ matrix.

$$ \left( \begin{array}{ccc} a & b \\ c & d \\ e & f \\ g & h \\ \end{array} \right)$$

The two constraints you mentioned give us two systems of equations.

$$ \left( \begin{array}{ccc} a & b \\ c & d \\ e & f \\ g & h \\ \end{array} \right) % \left( \begin{array}{ccc} 5 \\ 1 \end{array} \right) = \left( \begin{array}{ccc} 2 \\ 5 \\ 1 \\ 1 \end{array} \right) $$

$$\left( \begin{array}{ccc} a & b \\ c & d \\ e & f \\ g & h \\ \end{array} \right) % \left( \begin{array}{ccc} 1 \\ 0 \end{array} \right) = \left( \begin{array}{ccc} 3 \\ 4 \\ 2 \\ 2 \end{array} \right) $$

In this case the second system makes the solution very simple. BaronTV's hints can make things easier in the general case where the solution isn't so obvious.

Answer 3

Consider a linear transformation $f: \mathbb{R}^2 \to \mathbb{R}^4$.

Following from its linearity we have

$$f(\lambda v) = \lambda f(v) \text{ and } f(v+w) =f(v) + f(w) \quad (\lambda \in \mathbb R) \ (v, w \in \mathbb{R}^2) \tag{1}\ . $$

Firstly, where do the standard basis elements $(1,0)$ and $(0,1)$ get mapped to?

From $(1) $ we can conclude that all other elements in the image $\mathrm{im} (f) $ are multiples of those.

In matrix notation, we write $$[f]^{E_2}_{E_4} = \pmatrix{a&b\\c&d\\e&f\\g&h} $$ where the first column is the image of the first basis element of $\mathbb R ^2$ and the second column of the second basis element. We say that $f$ is represented with respect to the standard bases of $\mathbb R^2$ and $\mathbb R ^4$.

Now, how would we go about finding the second column by using that $f$ is linear?