📚 Topic Summary
The dimension of a vector space is the number of vectors in a basis for that vector space. A basis is a set of linearly independent vectors that span the entire vector space. To find the dimension, you need to identify a basis. If a vector space only contains the zero vector, its dimension is 0. For other vector spaces, the dimension corresponds to the number of 'free variables' after you've row-reduced the matrix representing the vectors in the space. Understanding linear independence and span is key to mastering this concept.
🧠 Part A: Vocabulary
Match the terms with their correct definitions:
| Term |
Definition |
| 1. Vector Space |
A. The set of all possible linear combinations of a set of vectors. |
| 2. Basis |
B. A set of linearly independent vectors that span the entire vector space. |
| 3. Linear Independence |
C. A set of vectors where no vector can be written as a linear combination of the others. |
| 4. Span |
D. A set of objects that can be added together and multiplied by scalars. |
| 5. Dimension |
E. The number of vectors in a basis for a vector space. |
(Match the numbers 1-5 with the letters A-E)
✍️ Part B: Fill in the Blanks
The dimension of a vector space $V$ is the number of vectors in a ________ for $V$. A basis is a set of ________ independent vectors that ________ the entire vector space. If the only vector in V is the zero vector, then the dimension of V is ________.
🤔 Part C: Critical Thinking
Explain, in your own words, why the dimension of $\mathbb{R}^n$ is equal to $n$. Provide an example to support your explanation.