# Linear Maps Examples 1

Recall from the Linear Maps page that a linear map or linear transformation from the vector space $V$ to the vector space $W$ is a function $T : V \to W$ such that for all $u, v \in V$ and for all $a \in \mathbb{F}$ we have that $T(u + v) = T(u) + T(v)$ (additivity property) and $T(av) = aT(v)$ (homogeneity property).

We will now look at some example questions regarding linear maps.

## Example 1

**Prove that $V$ is a vector space over the field $\mathbb{F}$ such that $\mathrm{dim} (V) = 1$ and $T \in \mathcal L (V, V)$ then for all $v \in V$ there exists a $\lambda \in \mathbb{F}$ such that $T(v) = \lambda v$.**

Let $V$ be a vector space over the field $\mathbb{F}$ and suppose that $\mathrm{dim} (V) = 1$. Let $u$ be any nonzero vector in $V$. Then $\{ u \}$ is a basis of $V$, and so $T(u) \in V$ is a scalar multiple of $u$, say $T(u) = \lambda u$.

Now let $v \in V$ be any arbitrary vector. Then $v = au$ for some $a \in \mathbb{F}$. Hence:

(1)Hence $T(v) = \lambda v$ for all $v \in V$.

## Example 2

**Provide an example of a transformation that has the homogeneity property but not the additivity property.**

Let $T : \mathbb{R}^2 \to \mathbb{R}$ be defined by $T(x, y) = x^{1/3} y^{2/3}$. If $u, v \in \mathbb{R}^2$ such that $u = (x_1, y_1)$ and $v = (x_2, y_2)$ and $a \in \mathbb{F}$ then the homogeneity property holds:

(2)However the additivity property does not hold:

(3)