๐ 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 that a subset $W$ of a vector space $V$ is a subspace, you must show that it is non-empty, closed under addition (if $\mathbf{u}$ and $\mathbf{v}$ are in $W$, then $\mathbf{u} + \mathbf{v}$ is in $W$), and closed under scalar multiplication (if $\mathbf{u}$ is in $W$ and $c$ is a scalar, then $c\mathbf{u}$ is in $W$). These conditions ensure that $W$ satisfies all the vector space axioms.
This worksheet provides a way to test your understanding of these fundamental concepts. From vocabulary matching to critical thinking questions, this will help you assess your knowledge. Ready to test yourself? Let's go!
๐ง 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 set of objects that satisfy specific axioms, allowing vector addition and scalar multiplication. |
| 3. Closure under Addition |
C. If $\mathbf{u}$ and $\mathbf{v}$ are in $W$, then $\mathbf{u} + \mathbf{v}$ is in $W$. |
| 4. Closure under Scalar Multiplication |
D. If $\mathbf{u}$ is in $W$ and $c$ is a scalar, then $c\mathbf{u}$ is in $W$. |
| 5. Span |
E. The set of all linear combinations of a set of vectors. |
๐ Part B: Fill in the Blanks
Complete the following paragraph using the words: vector space, subspace, zero vector, linear combination, and scalar multiplication.
A _________ is a subset of a larger _________. To verify that a subset is indeed a _________, one must show it contains the _________ and is closed under addition and _________. The span of a set of vectors is the set of all possible _________ of those vectors.
๐ค Part C: Critical Thinking
Explain, in your own words, why it is important to check for closure under addition and scalar multiplication when determining if a subset is a subspace.
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Answer Key
๐ง Part A: Vocabulary
- ๐ 1. Vector Space - B
- ๐ก 2. Subspace - A
- ๐ 3. Closure under Addition - C
- โ 4. Closure under Scalar Multiplication - D
- ๐ฑ 5. Span - E
๐ Part B: Fill in the Blanks
A subspace is a subset of a larger vector space. To verify that a subset is indeed a subspace, one must show it contains the zero vector and is closed under addition and scalar multiplication. The span of a set of vectors is the set of all possible linear combination of those vectors.
๐ค Part C: Critical Thinking
Closure under addition and scalar multiplication are critical because they ensure that the subset behaves like a vector space itself. If these properties do not hold, then vector addition or scalar multiplication could result in vectors that fall outside the subset, meaning the subset would not be a self-contained vector space and thus, not a subspace.