Podcast Beta
Questions and Answers
True or false: W = {(x, y) : x + 2y = 1} is a subspace of R^2?
True
True or false: W = {(x1, x2) : x1 >= 0 and x2 >= 0} is a subspace of R^2?
False
True or false: v = (1, 0) belongs to W?
True
True or false: (-1)v = (-1, 0) belongs to W?
Signup and view all the answers
True or false: W is closed under scalar multiplication?
Signup and view all the answers
True or false: The zero vector (0, 0) belongs to W?
Signup and view all the answers
True or false: W is a subspace of R^2?
Signup and view all the answers
True or false: W is closed under addition?
Signup and view all the answers
True or false: The subset W = {(x1, x2) : x1 >= 0 and x2 >= 0} is a proper subspace of R^2?
Signup and view all the answers
True or false: W is closed under scalar multiplication?
Signup and view all the answers
Study Notes
Subspaces
- A subspace is a nonempty subset W of a vector space V that is also a vector space with the same operations of addition and scalar multiplication.
- A subspace W of a vector space V must satisfy the following conditions:
- W is nonempty
- W is closed under addition
- W is closed under scalar multiplication
- W satisfies the ten axioms of a vector space
Example 1: W = {(x, y) | x + 2y = 1}
- This is not a subspace of R² because it does not contain the zero vector (0, 0).
- Consider v = (1, 0) ∈ W. (-1)v = (-1, 0) ∉ W, so W is not closed under scalar multiplication.
Example 2: W = {(x₁, x₂) | x₁ ≥ 0 and x₂ ≥ 0}
- This is not a subspace of R² because it is not closed under scalar multiplication.
- Let u = (1, 1) ∈ W. (-1)u = (-1, -1) ∉ W.
Trivial Subspaces
- Every vector space V has at least two subspaces: the zero vector space {0} and V itself.
- These two subspaces are called trivial subspaces.
- Any subspaces other than these two are called proper (or nontrivial) subspaces.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz tests your understanding of subspaces. It includes examples of determining whether a given set is a subspace or not.
