## Definition

We got a glimpse of linear transformations when we looked at the correspondence between elementary row operations and elementary matrices. We now look at linear transformations more generally.

Let $$V$$ and $$W$$ be vector spaces with scalars coming from the same field $$\mathbb{F}$$. A mapping $$T:V\rightarrow W$$ is a linear transformation if for any two vectors $$x$$ and $$y$$ in $$V$$ and any scalar $$\alpha \in \mathbb{F}$$, the following are satisfied: \begin{eqnarray} T(x+y) & = & T(x) + T(y) \\ T(\alpha x) & = & \alpha T(x) \end{eqnarray}

A couple of consequences of this definition are:

1. $$T(\alpha_1 x_1 + \cdots + \alpha_m x_m) = \alpha_1 T(x_1) + \cdots + \alpha_m T(x_m)$$ for any vectors $$x_1,\ldots, x_m \in V$$ and scalars $$\alpha_1,\ldots,\alpha_m \in \mathbb{F}$$.

2. $$T(0_V) = 0_W$$ where $$0_V$$ denotes the zero vector in $$V$$ and $$0_W$$ denotes the zero vector in $$W$$.

### Examples

1. Let $$A \in \mathbb{F}^{m \times n}$$. Then $$T(x) = Ax$$ is a linear transformation from $$\mathbb{F}^n$$ to $$\mathbb{F}^m$$. To see this, we need to check the two conditions. Let $$x,y \in \mathbb{F}^n$$ and $$\alpha \in \mathbb{F}$$. Then $$T(x+y) = A(x+y) = Ax + Ay = T(x) + T(y)$$ and $$T(\alpha x) = A(\alpha x) = \alpha (Ax) = \alpha T(x)$$.

2. Let $$T$$ be a mapping from $$P_2$$, the vector space of polynomials in $$x$$ with real coefficients having degree at most $$2$$, to $$\mathbb{R}^2$$ given by $$T(ax^2 + bx + c) = \begin{bmatrix} a+3c \\ a\end{bmatrix}$$. For example, $$T(2x^2 + 1) = \begin{bmatrix} 2 + 3(1) \\ 2\end{bmatrix} =\begin{bmatrix} 5 \\2 \end{bmatrix}$$ since $$2x^2 + 1$$ can be written as $$2 x^2 + 0x + 1$$.

To check that $$T$$ is a linear transformation, take two vectors $$a_1x^2+b_1x+c_1$$ and $$a_2x^2+b_2x+c_2$$ from $$P_2$$ and a real number $$\gamma$$. Then, \begin{eqnarray} & & T( (a_1x^2+b_1x+c_1) + (a_2x^2+b_2+c_2) ) \\ & = & T( (a_1+a_2)x^2+(b_1+b_2)x+(c_1+c_2) ) \\ & = & \begin{bmatrix} (a_1+a_2)+3(c_1+c_2) \\ a_1+a_2 \end{bmatrix} \\ & = & \begin{bmatrix} (a_1+3c_1)+(a_2+3c_2) \\ a_1+a_2 \end{bmatrix} \\ & = & \begin{bmatrix} a_1+3c_1 \\ a_1 \end{bmatrix} + \begin{bmatrix} a_2+3c_2 \\ a_2 \end{bmatrix} \\ & = & T(a_1x^2+b_1x+c_1) + T(a_2x^2+b_2 x+c_2), \end{eqnarray} and \begin{eqnarray} & & T( \gamma(a_1x^2+b_1x+c_1) ) \\ & = & T( (\gamma a_1)x^2+(\gamma b_1)x+\gamma c_1 ) \\ & = & \begin{bmatrix} \gamma a_1+3(\gamma c_1) \\ \gamma a_1 \end{bmatrix} \\ & = & \gamma\begin{bmatrix} a_1+3(c_1) \\ a_1 \end{bmatrix} \\ & = & \gamma T(a_1x^2+b_1x+c_1). \end{eqnarray}

## Exercises

1. Let $$T:\mathbb{R}^5\rightarrow \mathbb{R}$$ be such that $$T(x) = 0$$ for all $$x \in \mathbb{R}^5$$. Is $$T$$ a linear transformation? Explain your answer.

2. Let $$T:\mathbb{R}^3 \rightarrow \mathbb{R}^2$$ be a linear transformation such that $$T\left(\begin{bmatrix} 1 \\ 0 \\ 2\end{bmatrix}\right)= \begin{bmatrix} 1 \\ -4 \end{bmatrix}$$ and $$T\left(\begin{bmatrix} -1 \\ 1 \\ 1\end{bmatrix}\right)= \begin{bmatrix} 0 \\ 2 \end{bmatrix}$$. What is $$T\left(\begin{bmatrix} 4 \\ -1 \\ 5\end{bmatrix}\right)$$?