Linear Transformations

Why

Many functions of interest are additive and homogenous.

Definition

A transformation is linear (a linear transformation, linear map) if the result of a linear combination of the two vectors is the linear combination of the results of the vectors (using the same coefficients). The transformation is linear with respect to the field of the two vector spaces.

We use the term transformation (see Transformations) to emphasize and remind that the function is defined on a vector space. Of course, $\R $ is a vector space and so a function $f: \R \to \R $ may be linear. The linear maps from $\R $ to $\R $ are the the linear functions (see Real Linear Functions).

Notation

Let $(V, \F )$ and $(W, \F )$ be two vector spaces over the same field. Suppose $T: V \to W$. $T$ is linear means

\[ T(\alpha u + \beta v) = \beta T(u) + \alpha T(v) \quad \text{for all } \alpha , \beta \in F \text{ and } u, v \in V. \]

As usual, the condition that $T$ is linear condition is equivalent to the two conditions:

1. $T(u + v) = f(u) + T(v)$ for all $u, v \in V$, and
2. $f(\lambda u) = \lambda f(u)$ for all $\lambda \in \F $ and $u \in V$.

For linear maps, it is common to denote $T(v)$ by $Tv$; notice that we have dropped the usual parentheses.

We denote the set of all linear maps by $\mathcal{L} (V, W)$. It is understood when using this notation that $V$ and $W$ are vector spaces with respect to the same field $\F $.

Examples

Throughout, we consider vector spaces $V$ and $W$ over some fixed field $\F $.

Constant zero map. The map $T \in \mathcal{L} (V, W)$ defined by

\[ T(v) = 0 \in W \quad \text{for all } v \in V \]

The identity map. The map $T \in \mathcal{L} (V, V)$ defined by

\[ Tv = v \quad \text{for all } v \in V \]

Differentiation of polynomials Suppose $P$ is the set of all polynomials with coefficients in $\R $. (Some authors denote this set by $\mathcal{P} (\R )$. Recall that every $p \in \mathcal{P} (\R )$ is differentiable and $p' \in \mathcal{P} (\R )$. The map $T \in \mathcal{L} (\mathcal{P} (\R ), \mathcal{P} (\R ))$ defined by

\[ Tp = p' \]

Integration of polynomials As in the previous paragraph, $\mathcal{P} (\R )$ denotes the vector space of polynomials with coefficients in $\R $. The map $T \in \mathcal{L} (\mathcal{P} (\R ), \R )$ defined by

\[ Tp = \int _{[0,1]} p \]

Multipliciation by a quadratic. As in the previous paragraph, $\mathcal{P} (\R )$ denotes the vector space of polynomials with coefficients in $\R $. The map $T \in \mathcal{L} (\mathcal{P} (\R ), \mathcal{P} (\R ))$ defined by

\[ (Tp)(x) = x^2p(x) \quad \text{for all } x \in \R , p \in \mathcal{P} (\R ) \]

Sequence backward shift. Denote the space of infinite sequences in a field $\F $ by $\F ^\N $ as usual. Define $T \in \mathcal{L} (\F ^\N , \F ^\N )$ by

\[ T(x_1, x_2, x_3, \dots ) = (x_2, x_3, \dots ) \]

From real space the the real plane. Define $T \in \mathcal{L} (\R ^3, \R ^2)$ by

\[ T(x, y, z) = (2x - y + 3z, 7x+5y - 6z) \]

From $\F ^n$ to $\F ^m$. Generalizing the previous example, suppose $m$ and $n$ are natuarl numbers, and let $A_{i, j} \in \F $ for $i = 1, \dots , m$ and $j = 1, \dots , m$. Define $T \in \mathcal{L} (\F ^3, \F ^2)$ by

\[ T(x_1, \cdots, x_n) = (A_{1,1}x + \cdots + A_{1,n} x_n, \dots , A_{m,1}x_1 + \cdots + A_{m,n}x_n) \]

A counterexample: $\cos$¹ Notice $\cos(x + y) = \cos(x) + \cos(y)$. True, $\cos$ is not homogenous. that $\cos2x = 2\cos(x)$ and But this does not hold for all reals: $\cos \lambda x \neq \lambda \cos(x)$.

1. Need to add a sheet for trigonometric functions. ↩︎