Linear Transformations

Definition. A function $T: V \to W$ between two $k$-vector spaces $V$ and $W$ is a linear transformation if

• $T$ commutes with scalar multiplication: \(T(cv) = cT(v)\) for all $c \in k$ and all $v \in V$ and
• $T$ commutes with addition of vectors: \(T(v_1 + v_2) = T(v_1) + T(v_2)\) for all $v_1,v_2 \in V$.

As indicated by the example below, you have seen many linear transformations in your life already and they all do not arise a priori as multiplication by a matrix.

Linear transformations provide the right abstraction for drawing conclusions as we would for matrix multiplication but in a broader context.

Examples

• For any vector space $V$, the identity map \(\begin{aligned} \operatorname{Id}_V : V & \to V \\ v & \mapsto v \end{aligned}\) is a linear transformation.

• Multiplication by a matrix is a linear transformation. We know from matrix algebra that \(A(cx) = cAx\) and that \(A(x + y) = Ax + Ay\) for vectors $x,y$. But we can also multiply matrices by matrices.

  Take the vector space $\operatorname{Mat}_{m,n}(k)$ of $m \times n$ matrices with entries in a field $k$ and let $A$ be a $m \times m$ matrix. Then, we get a function \(\begin{aligned} \operatorname{Mat}_{m,n}(k) & \to \operatorname{Mat}_{m,n}(k) \\ B & \mapsto AB \end{aligned}\) This is also a linear transformation. Again, we know that scalar multiplication commutes with matrix multiplication so \(A(cB) = cAB\) and we know that we can distribute multiplication over addition of matrices \(A(B+C) = AB + AC\) Thus, we have a linear transformation.

• Transpose is a linear transformation. We have \((cA)^T = cA^T\) and \((A+B)^T = A^T + B^T\)

• Differentiation is a linear transformation. We recall from calculus that \(\frac{d}{dx} (cf) = c \frac{df}{dx}\) and \(\frac{d}{dx}(f + g) = \frac{df}{dx} + \frac{dg}{dx}\)

• Integration is a linear transformation. Again from calculus we know that \(\int cf(x) \ dx = c \int f(x) \ dx\) and \(\int f(x) + g(x) \ dx = \int f(x) \ dx + \int g(x) \ dx.\)

• If we have a bilinear pairing $\langle -,- \rangle$ on a vector space $V$ and a vector $v \in V$, we can make a function \(\begin{aligned} V & \to k \\ w & \mapsto \langle v,w \rangle \end{aligned}\) This is a linear transformation since \(\langle v, c_1w_1 + c_2w_2 \rangle = c_1 \langle v,w_1 \rangle + c_2 \langle v,w_2 \rangle\) Supping this up a little, given a subspace $U \subseteq V$, then the projection \(\operatorname{proj}_U : V \to U\) is a linear transformation.

• More geometrically, rotations and reflections in the plane are linear transformations. Try to visualize this.

• For something that is not a linear transformation, take a look at \(\begin{aligned} \mathbb{R} & \to \mathbb{R} \\ x & \mapsto x^{10} \end{aligned}\)
  We don’t preserve scaling since \((5x)^{10} = 5^{10} x^{10} \neq 5 x^{10}\) for $x \neq 0$.

• Another, more physical, example of a nonlinear transformation is falling. If $h(t)$ is the height off the ground after $t$ seconds of falling from a plane, then $h(t)$ is a not linear function. For a linear function, the distance you fall from time $t = 0$ to $t=1$ is the same as that from $t=1$ to $t=2$. But, due to gravity’s acceleration, you speed up and fall faster the longer you fall.

Properties

Lemma. The following statements are equivalent for a function $T: V \to W$ between $k$-vector spaces $V$ and $W$.

• $T$ is a linear transformation,
• $T$ preserves linear combinations: \(T\left( \sum c_i v_i \right) = \sum c_iT(v_i)\)
• $T$ preserves linear combinations of two vectors.

The next lemma can provide a quick sanity check for whether a functor could possibly be a linear transformation.

Lemma. If $T$ is a linear transformation, the $T(0) = 0$. In particular, functions that do not take the zero vector to the zero vector cannot be linear transformations.