Kernel and Image
Chapter 7 • Section 7-2
"Where vectors go to die (zero), and where they can live. 💀🌍"
🎯 Core Concepts
Kernel (Null Space)
The set of all vectors in $V$ that map to the zero vector in $W$.
Corresponds to solutions of $A\vec{x}=\vec{0}$.
Image (Range)
The set of all possible output vectors in $W$.
Corresponds to the column space of $A$.
📜 Key Theorems
Theorem: Subspaces
- $\ker(T)$ is a subspace of the domain $V$.
- $\im(T)$ is a subspace of the codomain $W$.
This means they contain the zero vector and are closed under addition/scaling.
Example: Polynomial Evaluation
Let $T: \calP_1 \to \mathbb{R}$ be defined by $T(p(x)) = p(1)$.
Kernel:
Polynomials where $p(1)=0$.
$\ker(T) = \{ a(x-1) \mid a \in \mathbb{R} \}$.
Image:
All possible values of $p(1)$.
$\im(T) = \mathbb{R}$ (since we can get any value).
Theorem: Injectivity
A linear transformation $T$ is one-to-one if and only if:
If only the zero vector maps to zero, then no two other vectors map to the same place!
Rank-Nullity Visualizer
The Rank-Nullity Theorem states: $\dim(V) = \rank(T) + \nullity(T)$.
Adjust the slider to see how Rank and Nullity trade off for a fixed dimension $n$.
Rank (3) + Nullity (2) = 5
Types of Mappings
One-to-One (Injective)
Distinct inputs map to distinct outputs.
$\ker(T) = \{\vec{0}\}$
Nullity is 0. Full column rank.
Onto (Surjective)
Every vector in $W$ is hit by at least one input.
$\im(T) = W$
Rank equals $\dim(W)$. Full row rank.
🧠 Knowledge Check Win $15
If $T: \mathbb{R}^5 \to \mathbb{R}^3$ is onto, what is the dimension of its kernel (nullity)?
Hint: Use Rank-Nullity Theorem. $n=5$, Rank=3 (since onto $\mathbb{R}^3$).