An equivalent definition of a linear map $$T$$ from $$V$$ to $$W$$ is a function $T : V \to W$ such that for all vectors $u, v \in V$ , $$T(u + v) = T(u) + T(v)$$ (Additivity) and for all $a \in \mathbb{F}$ , $$T(au) = aT(u)$$ (Homogeneity).

We will now look at some examples of linear maps and verify that they in fact a linear maps by showing that $$T(au + bv) = aT(u) + bT(v)$$ for all $u, v \in V$ and $a, b \in \mathbb{F}$ .

The Zero Map

The zero map denoted $0 \in \mathcal L (V, W)$ defined by $$0(v) = 0$$ maps every vector $v \in V$ to the zero vector $0_W \in W$ .

Let $u, v \in V$ and $a, b \in \mathbb{F}$ .

$$0(au + bv) = 0_W$$

$$a 0(u) + b 0(v) = 0_W + 0_W = 0_W$$ .

Therefore $$0(au + bv) = a0(u) + b0(v)$$ , so we have verified that $$0$$ is a linear map.

The Identity Map

The identity map denoted $I \in \mathcal L (V, V)$ defined by $$I(v) = v$$ maps every vector $v \in V$ back to itself.

Let $u, v \in V$ and $a, b \in \mathbb{F}$ .

$$I(au + bv) = au + bv$$

$$aI(u) + bI(v) = au + bv$$ .

Therefore $$I(au + bv) = aI(u) + bI(v)$$ , so we have verified that $$I$$ is a linear map.

The Differentiation of Polynomials Map

The differentiation of polynomials map $T \in \mathcal L (\wp (\mathbb{R}), \wp (\mathbb{R}))$ defined by $$T(p(x)) = p'(x)$$ takes every vector $p(x) \in \wp (\mathbb{R})$ and maps it to its derivative $$p'(x)$$ .

Let $p(x), q(x) \in \wp ( \mathbb{R} )$ and $a, b \in \mathbb{F}$ .

$$T(a p(x) + b q(x)) = (ap(x) + bq(x))' = ap'(x) + bq'(x)$$

$$aT(p(x)) + bT(q(x)) = ap'(x) + bp'(x)$$ .

Therefore $$T(ap(x) + bq(x)) = aT(p(x)) + bT(q(x))$$ , and so we have verified that $$T$$ is a linear map.

The Definite Integration of Polynomials Map

The integration of polynomials map from $$c$$ to $$d$$ ( $a, b \in \mathbb{R}$ ) denoted $T \in \mathcal L (\wp (\mathbb{R}), \mathbb{R} )$ defined by $T(p(x)) = \int_c^d p(x) \: dx$ maps vector $p(x) \in \wp (\mathbb{R})$ to a number in $\mathbb{R}$ .

Let $p(x), q(x) \in \wp ( \mathbb{R} )$ and $a, b \in \mathbb{F}$ .

$T(a p(x) + b q(x)) = \int_c^d ap(x) + bq(x) \: dx = a \int_c^d p(x) \: dx + b \int_c^d q(x) \: dx$ .

$aT(p(x)) + bT(q(x)) = a \int_c^d p(x) \: dx + b \int_c^d q(x) \: dx$ .

Therefore $$T(ap(x) + bq(x)) = aT(p(x)) + bT(q(x))$$ , and so we have verified that $$T$$ is a linear map.

The Left Shift Operator

The left shift operator $T \in \mathcal L (\mathbb{F}^{\infty}, \mathbb{F}^{\infty})$ defined by $$T((x_1, x_2, ...)) = (x_2, x_3, ...)$$ takes every sequence in $\mathbb{F}^{\infty}$ and shifts every term one position to the left.

Let $(x_1, x_2, ...), (y_1, y_2, ...) \in \mathbb{F}^{\infty}$ and $a, b \in \mathbb{F}$ .

$$T(a(x_1, x_2, ...) + b(y_1, y_2, ...)) = T((ax_1 + by_1, ax_2 + by_2, ...)) = (ax_2 + by_2, ax_3 + by_3, ...)$$ .

$$aT((x_1, x_2, ...)) + bT((y_1, y_2, ...)) = a(x_2, x_3, ...) + b(y_2, y_3, ...) = (ax_2, ax_3, ...) + (by_2, by_3, ...) = (ax_2 + by_2, ax_3 + by_3, ...)$$ .

Therefore $$T(a(x_1, x_2, ...) + b(y_1, y_2, ...)) = aT((x_1, x_2, ...)) + bT((y_1, y_2, ...))$$ , and so we have verified that $$T$$ is a linear map.

The Right Shift Operator

The right shift operator $T \in \mathcal L (\mathbb{F}^{\infty}, \mathbb{F}^{\infty})$ defined by $$T((x_1, x_2, ...)) = (0, x_1, ...)$$ takes every sequence in $\mathbb{F}^{\infty}$ and shifts every term one position to the right while replacing the first term with zero.