📚 Topic Summary
The Spanning Set Theorem is a cornerstone of linear algebra, providing powerful tools for understanding vector spaces and their subspaces. In essence, it allows us to determine when a vector can be removed from a spanning set without changing the span, and when a vector can be added to a linearly independent set without losing linear independence. This is crucial for finding minimal spanning sets (bases) and understanding the dimension of a vector space.
Specifically, the theorem has two main parts. First, if a vector $v$ in a spanning set $S$ can be written as a linear combination of the other vectors in $S$, then removing $v$ from $S$ does not change the span of $S$. Second, if we have a linearly independent set and add a vector that is not in the span of that set, the new set is also linearly independent.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Spanning Set |
A. A set of vectors where no vector can be written as a linear combination of the others. |
| 2. Linear Combination |
B. A set of vectors whose span is the entire vector space. |
| 3. Linear Independence |
C. A set of vectors that, when multiplied by scalars and added together, produce another vector. |
| 4. Vector Space |
D. The set of all possible linear combinations of a set of vectors. |
| 5. Span |
E. A set that is closed under scalar multiplication and vector addition. |
(Answers: 1-B, 2-C, 3-A, 4-E, 5-D)
📝 Part B: Fill in the Blanks
Complete the following paragraph using the words provided: span, linearly dependent, vector, spanning, linear combination.
The Spanning Set Theorem helps us understand when we can remove a ______ from a ______ set without changing the ______ of the set. If a vector can be written as a ______ of the other vectors in the set, then the set is ______.
(Answers: vector, spanning, span, linear combination, linearly dependent)
💡 Part C: Critical Thinking
Explain, in your own words, why the Spanning Set Theorem is useful for finding a basis for a vector space.