📚 Topic Summary
In linear algebra, a subspace is a subset of a vector space that is itself a vector space. The dimension of a subspace is the number of vectors in a basis for that subspace. Understanding dimension is key because it tells us how many independent vectors are needed to span the entire subspace. Different subspaces of the same vector space can have different dimensions. For example, in $\mathbb{R}^3$, a line through the origin is a subspace of dimension 1, a plane through the origin is a subspace of dimension 2, and the entire space $\mathbb{R}^3$ itself is a subspace of dimension 3.
When working with subspaces, remember to verify that the subset satisfies the conditions to be a subspace (closure under addition and scalar multiplication). Then, find a basis for the subspace to determine its dimension.
🧮 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Basis |
A. A subspace that contains only the zero vector. |
| 2. Dimension |
B. The number of vectors in a basis of a vector space. |
| 3. Span |
C. A set of vectors that generates the entire vector space. |
| 4. Trivial Subspace |
D. The set of all linear combinations of a set of vectors. |
| 5. Vector Space |
E. A set that is closed under addition and scalar multiplication. |
Matching Answers: 1-C, 2-B, 3-D, 4-A, 5-E
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct words.
The ________ of a subspace $W$ is the number of vectors in any ________ for $W$. If $W$ is the trivial subspace, then the dimension of $W$ is ________. The dimension of the vector space $\mathbb{R}^n$ is ________. To find a basis for a subspace, determine a set of linearly ________ vectors that span the subspace.
Answer: dimension, basis, zero, n, independent
🤔 Part C: Critical Thinking
Suppose $V$ is a vector space with dimension $n$. If $W$ is a subspace of $V$, what can you say about the dimension of $W$?
Answer: The dimension of $W$ must be less than or equal to $n$. In other words, dim($W$) $\leq$ n.