Podcast
Questions and Answers
True or false: W = {(x, y) : x + 2y = 1} is a subspace of R^2?
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?
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 or false: v = (1, 0) belongs to W?
True
True or false: (-1)v = (-1, 0) belongs to W?
True or false: (-1)v = (-1, 0) belongs to W?
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True or false: W is closed under scalar multiplication?
True or false: W is closed under scalar multiplication?
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True or false: The zero vector (0, 0) belongs to W?
True or false: The zero vector (0, 0) belongs to W?
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True or false: W is a subspace of R^2?
True or false: W is a subspace of R^2?
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True or false: W is closed under addition?
True or false: W is closed under addition?
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True or false: The subset W = {(x1, x2) : x1 >= 0 and x2 >= 0} is a proper subspace of R^2?
True or false: The subset W = {(x1, x2) : x1 >= 0 and x2 >= 0} is a proper subspace of R^2?
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True or false: W is closed under scalar multiplication?
True or false: W is closed under scalar multiplication?
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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.
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Description
This quiz tests your understanding of subspaces. It includes examples of determining whether a given set is a subspace or not.
