Subspace Quiz

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Questions and Answers

True or false: W = {(x, y) : x + 2y = 1} is a subspace of R^2?

True (A)

True or false: W = {(x1, x2) : x1 >= 0 and x2 >= 0} is a subspace of R^2?

False (B)

True or false: v = (1, 0) belongs to W?

True (A)

True or false: (-1)v = (-1, 0) belongs to W?

False (B) Signup and view all the answers

True or false: W is closed under scalar multiplication?

False (B) Signup and view all the answers

True or false: The zero vector (0, 0) belongs to W?

False (B) Signup and view all the answers

True or false: W is a subspace of R^2?

False (B) Signup and view all the answers

True or false: W is closed under addition?

False (B) 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?

True (A) Signup and view all the answers

True or false: W is closed under scalar multiplication?

False (B) Signup and view all the answers

Flashcards

Is W a subspace?

False. A subspace must contain the zero vector, be closed under scalar multiplication, and closed under vector addition. The set W = {(x1, x2) : x1 >= 0 and x2 >= 0} isn't closed under scalar multiplication.

Is W closed under scalar multiplication?

False. For W to be a subspace of R^2, multiplying a vector in W by any scalar must result in a vector that is also in W. However, if v = (1, 0) is in W, (-1)v = (-1, 0) is not in W.

Does the zero vector belong to W?

False. For W to be a subspace, it must contain the zero vector (0, 0). In this case, (0, 0) does not satisfy the condition x1 >= 0 and x2 >= 0.

Is W closed under addition?

False. For W to be closed under addition, the sum of any two vectors in W must also be in W. W = {(x1, x2) : x1 >= 0 and x2 >= 0}.

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Is subset W a proper subspace of R^2?

True. The subset W = {(x1, x2) : x1 >= 0 and x2 >= 0} is not a subspace of R^2, however, it is a proper subset.

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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.