Linear Maps Examples 3

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 more example questions regarding linear maps.

Example 1

Let $T \in \mathcal L (V, W)$ and suppose that $\{ T(v_1), T(v_2), ..., T(v_m) \}$ is a set of linearly independent vectors in $W$. Prove that $\{ v_1, v_2, ..., v_m \}$ is a set of linearly independent vectors in $V$.

Let $T \in \mathcal L (V, W)$ and suppose that $\{ T(v_1) T(v_2), ..., T(v_m) \}$ is a set of linearly independent vectors in $W$.

Consider the following vector equation for $a_1, a_2, ..., a_m \in \mathbb{F}$:

(1)

\begin{align} \quad a_1v_1 + a_2v_2 + ... + a_mv_m = 0 \end{align}

Now apply the linear transformation $T$ to both sides of the equation above to get that:

(2)

\begin{align} \quad T(a_1v_1 + a_2v_2 + ... + a_mv_m) = T(0) \\ \quad a_1T(v_1) + a_2T(v_2) + ... + a_mT(v_m) = 0 \end{align}

The equation above implies that $a_1 = a_2 = ... = a_m = 0$ since $\{ T(v_1), T(v_2), ..., T(v_m) \}$ is a set of linearly independent vectors in $W$. Hence $\{ v_1, v_2, ..., v_m \}$ is a set of linearly independent vectors in $V$.

Example 2

Consider the linear map $T : \wp (\mathbb{R}) \to \mathbb{R}^2$ defined by $T(p(x)) = \left (p(1) + 2p(2) + 3p(3), \int_0^1 x^2 p(x) \: dx \right )$. Determine whether or not $T$ is a linear map from $\wp (\mathbb{R})$ to $\mathbb{R}^2$.

We will first show that the additivity property holds. Let $p(x), q(x) \in \wp (\mathbb{R})$. Then:

(3)

\begin{align} \quad T(p(x) + q(x)) = \left ( [p(1) + q(1)] + 2[p(2) + q(2)] + 3[p(3) + q(3)], \int_0^1 x^2 [p(x) + q(x)] \: dx \right ) \\ \quad T(p(x) + q(x)) = \left ( p(1) + 2p(2) + 3p(3), \int_0^1 x^2 p(x) \: dx \right ) + \left ( q(1) + 2q(2) + 3q(3), \int_0^1 x^2 q(x) \: dx \right ) \\ \quad T(p(x) + q(x)) = T(p(x)) + T(q(x)) \end{align}

So the additivity property holds. We will now show that the homogeneity property holds. Let $p(x) \in \wp (\mathbb{R})$ and let $a \in \mathbb{F}$. Then:

(4)

\begin{align} \quad T(ap(x)) = \left ( ap(1) + 2ap(2) + 3ap(3), \int_0^1 x^2 [ap(x)] \: dx \right ) \\ \quad T(ap(x)) = \left ( a[p(1) + 2p(2) + 3p(3)], a \int_0^1 x^2 p(x) \: dx \right ) \\ \quad T(ap(x)) = a \left (p(1) + 2p(2) + 3p(3), \int_0^1 x^2 p(x) \: dx \right ) \\ \quad T(ap(x)) = aT(p(x)) \end{align}

Therefore the homogeneity property holds and so $T$ is indeed a linear map from $\wp (\mathbb{R})$ to $\mathbb{R}$.