Abstract linear transformations can exist between any vector spaces. These vector spaces of course include $\mathbb{R}^n$ but could also be

• $\mathbb{C}(\mathbb{R})$ - continuous single-variable functions $f: \mathbb{R} \rightarrow \mathbb{R}$
• $\mathbb{C}'(\mathbb{R})$ - continuous single-variable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ with continuous derivatives
• $\mathbb{P}(\mathbb{R})$ - all polynomials with real coefficients
• $\mathbb{P}_n(\mathbb{R})$ - polynomials with real coefficients and $\text{degree} \leq n$

A subpace of a vector space $V$ is a subset $U \subset V$ such that

1. $\vec{0} \in U$
2. $\vec{x}, \vec{y} \in U$ ⇒ $\vec{x} + \vec{y} \in U$
3. $\vec{x} \in U, \lambda \in \mathbb{R}$ ⇒ $\lambda \vec{x} \in U$

[Preserves Addition]

[Preserves Scalar Multiplication]

The kernel and range of a linear transformation $T: V \rightarrow W$ is defined as

$$ \text{Ker}(T) = \{\vec{x} \in V \text{ s.t. } T(\vec{x}) = \vec{0} \} \\ \text{Rng}(T) = \{T(\vec{x}) \in W \text{ s.t. } \vec{x} \in V \} $$

In this figure, you can see

• Kernel is everything on the left that maps to 0 on the right
• Range is everything on the right that gets mapped to

Fact: The kernel of $T: V \rightarrow W$ is a subspace of $V$, and the range of $T$ is a subspace of $W$

If we only consider linear transformations between $\mathbb{R}^n$ and $\mathbb{R}^m$ , then we know that these transformations can be written as $T_A(\vec{x}) = A\vec{x}$ where $A$ is the $m \times n$ standard matrix of $T_A$. With these linear transformations between n-tuples of real numbers, we have analogous concepts to kernel and range called the null-space and column-space of $A$:

$$ \text{Ker}(T_A) = \text{Nul}(A)\\ \text{Rng}(T_A) = \text{Col}(A) $$

Fact: $\text{Nul}(A) \subset \mathbb{R}^n$ is a subspace, and $Col(A) \subset \mathbb{R}^m$ is a subspace