# 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)

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

(2)

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)

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)

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

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