📚 Topic Summary
In linear algebra, a subspace is a subset of a vector space that is itself a vector space under the same operations defined for the larger vector space. To prove a subset $W$ is a subspace of a vector space $V$, you must show that $W$ is non-empty, closed under addition (i.e., if $\mathbf{u}, \mathbf{v} \in W$, then $\mathbf{u} + \mathbf{v} \in W$), and closed under scalar multiplication (i.e., if $\mathbf{u} \in W$ and $c$ is a scalar, then $c\mathbf{u} \in W$). These three conditions guarantee that $W$ satisfies all the vector space axioms. Proving these properties is the core skill tested by subspace proof worksheets.
Essentially, subspace proofs are about verifying that a smaller space behaves like the larger vector space it lives within. It's a key concept for understanding linear transformations and other advanced topics.
🧠 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Vector Space |
A. A subset of a vector space that is itself a vector space. |
| 2. Subspace |
B. A real number that scales a vector. |
| 3. Closure under Addition |
C. A set with operations of addition and scalar multiplication that satisfy certain axioms. |
| 4. Closure under Scalar Multiplication |
D. If $\mathbf{u}$ is in W, then $c\mathbf{u}$ is also in W. |
| 5. Scalar |
E. If $\mathbf{u}$ and $\mathbf{v}$ are in W, then $\mathbf{u} + \mathbf{v}$ is also in W. |
✏️ Part B: Fill in the Blanks
To prove that a subset $W$ of a vector space $V$ is a __________, you must show three things: first, that $W$ is __________. Second, that $W$ is closed under __________, meaning that if $\mathbf{u}$ and $\mathbf{v}$ are in $W$, then their __________ is also in $W$. Third, that $W$ is closed under __________, meaning that if $\mathbf{u}$ is in $W$ and $c$ is a scalar, then $c\mathbf{u}$ is also in $W$.
🤔 Part C: Critical Thinking
Let $V = \mathbb{R}^2$. Let $W = \{(x, y) \in \mathbb{R}^2 : x = y^2\}$. Is $W$ a subspace of $V$? Justify your answer with a proof or a counterexample.