What is Kernel and image?

Example 20.8 (Building a map to specification)

Construct $u \in \mathcal{L}(\R^3)$ with $\ker u = \operatorname{Vect}(1,1,1)$ and $\operatorname{im} u = \{z = 0\}$. Sanity first: rank–nullity demands $1 + 2 = 3$ — consistent, so a solution may exist. Choose a basis adapted to the kernel, say $\bigl((1,1,1),\ e_1,\ e_2\bigr)$ (Example 19.7), and prescribe images (Proposition 20.2):

Then $\ker u \supseteq \operatorname{Vect}(1,1,1)$ and $\operatorname{im} u = \operatorname{Vect}(e_1, e_2) = \{z = 0\}$; rank–nullity forces $\dim\ker u = 1$, so the kernel is exactly the prescribed line. Explicitly, decomposing $(x, y, z) = z(1,1,1) + (x - z)e_1 + (y - z)e_2$:

The recipe generalizes: a linear map with prescribed kernel $N$ and image $I$ exists exactly when $\dim N + \dim I = \dim E$ — necessity is rank–nullity, sufficiency is this construction.

Example 20.10 (Interpolation, structurally)

Fix distinct $x_0, \dots, x_n$ and let $u \colon \R_n[X] \to \R^{n+1}$, $P \mapsto (P(x_0), \dots, P(x_n))$: linear. Its kernel is $\{P : \deg P \leq n,\ n+1 \text{ roots}\} = \{0\}$ (Corollary 8.8). Equal dimensions $n + 1$: $u$ is an isomorphism — existence and uniqueness of the Lagrange interpolant (Theorem 8.23) in one line.

The same one-line pattern handles data mixing values and derivatives: $v \colon \R_3[X] \to \R^4$, $P \mapsto \bigl(P(0), P'(0), P(1), P'(1)\bigr)$ is linear, and its kernel consists of degree-$\leq 3$ polynomials with double roots at $0$ and $1$, i.e. divisible by $X^2(X-1)^2$ of degree $4$: only $P = 0$. Equal dimensions again: every quadruple of data $(P(0), P'(0), P(1), P'(1))$ is realized by exactly one cubic — Hermite interpolation, granted by a kernel computation before any formula is written (the weekend problem of Chapter 22 meets its determinant).

Example 20.11 (Rank–nullity at work: the difference operator)

Let $\Delta \colon \R_n[X] \to \R_n[X]$, $P \mapsto P(X+1) - P(X)$: linear. Kernel: if $\Delta P = 0$, then $P(0) = P(1) = P(2) = \dots$, so $P - P(0)$ has infinitely many roots and vanishes (Corollary 8.8): $\ker\Delta$ is the line of constants. Rank–nullity: $\operatorname{rk}\Delta = (n + 1) - 1 = n$. Since $\deg \Delta P < \deg P$ for nonconstant $P$ (the top terms cancel), $\operatorname{im}\Delta \subseteq \R_{n-1}[X]$, which has dimension exactly $n$: the inclusion is an equality. Conclusion, with no computation of preimages: every polynomial $Q$ of degree $\leq n - 1$ is a difference $Q = P(X+1) - P(X)$ — the discrete antiderivative exists. (Compare the weekend problem of Chapter 18, where $\Delta$ was inverted explicitly in the binomial basis.)