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)

\begin{align} \quad T(v) = T(au) = a T(u) = a (\lambda u) = \lambda (au) = \lambda v \end{align}

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)

\begin{align} \quad T(au) = T(ax_1, ay_1) = (ax_1)^{1/3} (ay_1)^{2/3} = a^{1/3}x_1^{1/3}a^{2/3}y_1^{2/3} = a(x^{1/3}y^{2/3}) = a T(u) \end{align}

However the additivity property does not hold:

(3)

\begin{align} \quad T(u + v) = T(x_1 + x_2, y_1 + y_2) = (x_1 + x_2)^{1/3} (y_1 + y_2)^{2/3} \neq x_1^{1/3}y_1^{2/3} + x_2^{1/3} y_2^{2/3} = T(u) + T(v) \end{align}