📚 Topic Summary
The dimension of a vector space $V$ is the number of vectors in a basis for $V$. A basis is a set of linearly independent vectors that span $V$. Think of it as the minimum number of coordinates you need to describe any vector in that space. If a vector space has a finite basis, it's called finite-dimensional; otherwise, it's infinite-dimensional. Understanding dimension helps us classify and compare different vector spaces.
🧠 Part A: Vocabulary
Match each term with its correct definition:
| Term |
Definition |
| 1. Vector Space |
A. The number of vectors in a basis. |
| 2. Basis |
B. A set of linearly independent vectors that span the vector space. |
| 3. Dimension |
C. A set that is closed under addition and scalar multiplication. |
| 4. Linear Independence |
D. A property where no vector in the set can be written as a linear combination of the others. |
| 5. Span |
E. The set of all possible linear combinations of a set of vectors. |
(Answers: 1-C, 2-B, 3-A, 4-D, 5-E)
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
The dimension of a vector space is the number of vectors in its _____. A _____ is a set of _____ independent vectors that _____ the entire vector space. If the only solution to $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ is $c_1 = c_2 = ... = c_n = 0$, then the vectors $v_1, v_2, ..., v_n$ are considered _____.
(Answers: basis, basis, linearly, span, linearly independent)
💡 Part C: Critical Thinking
Explain, in your own words, why understanding the dimension of a vector space is important in linear algebra. Provide an example of how it can be used.