# Procedure to convert matrix representation into a linear transfer function

A linear transform $$T : R^3 \rightarrow R^2$$ has the matrix representation:

$$T = \begin{bmatrix} 2 & -1 & 2 \\ 4 & 1 & 5 \end{bmatrix}$$

the $$R^2$$ basis is (4,3), and (3,2).

the $$R^3$$ basis is the standard basis: (1,0,0), (0,1,0), (0,0,1)

What's the procedure to convert matrix T into a linear transform function T(a,b,c)?

$$T(a,b,c) = a T(1,0,0) + b T(0,1,0) + c T(0,0,1)$$

$$T(a,b,c) = (20a - b + 23c, 14a -b + 16c)$$

Let $$B$$ be that basis of $$\mathbb{R}^2$$.

First, you compute $$T(1,0,0)$$, $$T(0,1,0)$$, and $$T(0,0,1)$$. These are equal to $$(2,4)_B$$, to $$(-1,1)_B$$ and to $$(2,5)_B$$. But

• $$(2,4)_B=(20,14)$$;
• $$(-1,1)_B=(-1,-1)$$;
• $$(2,5)_B=(23,16)$$.

So\begin{align}T(a,b,c)&=aT(1,0,0)+bT(0,1,0)+cT(0,0,1)\\&=a(20,14)+b(-1,-1)+c(23,16)\\&=(20a-b+23c,14a-b+16c).\end{align}So, yes, that answer from your book is indeed correct.

$$T: V \rightarrow W$$

T Matrix Representation: $$T = \begin{bmatrix} 2 & -1 & 2 \\ 4 & 1 & 5 \end{bmatrix}$$

Basis Sets:

$$basis\{V\} = \left\{v_1=\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}, v_2=\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}, v_3=\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right\}$$

$$basis\{W\} = \left\{w_1=\begin{bmatrix}4 \\ 3 \end{bmatrix}, w_2=\begin{bmatrix}3 \\2 \end{bmatrix}\right\}$$

A = matrix formed by concatenating the column vectors of V basis

B = matrix formed by concatenating the column vectors of W basis

$$B=\begin{bmatrix}4 & 3 \\ 3 & 2\end{bmatrix}$$

\begin{aligned} T(v_1) &= B\ col(1, T) \\ T(v_2) &= B\ col(2, T) \\ T(v_3) &= B\ col(3, T) \end{aligned}

$$v_x = \begin{bmatrix} a\\ b\\ c \end{bmatrix}$$

$$T(v_x) = aT(v_1) +bT(v_2)+cT(v_3)$$

or:

$$T(a,b,c) = aT(v_1) +bT(v_2)+cT(v_3)$$