Linear Algebra: Vector Subspaces

In summary, the union of two vector subspaces is not always a subspace. A counter-example to this statement is the union of the sets {(x, 0)} and {(0, y)} in R2, which does not contain the vector (2, 3) and thus is not closed under addition. Therefore, the statement is false.
  • #1
war485
92
0

Homework Statement



True/false: Union of two vector subspaces is a subspace.

Homework Equations



none

The Attempt at a Solution



I'm unsure if this is true because I'm also unsure if it already assumes that it is closed under scalar multiplication and addition. If it is closed, then I'd like to think it is true, but it might be false if it isn't closed. Help me a lil' to know what's already assumed?
 
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  • #2
Generally that is false, the union of two vector subspaces is not a subspace. To prove it is false, first assume that it is true and look for a counter-example. Let me see what you can come up with.
 
  • #3
how about span of e1 and span of e2. Then their union is not closed under vector addition, then the union is not a subspace, woah.
 
  • #4
war485 said:

Homework Statement



True/false: Union of two vector subspaces is a subspace.

Homework Equations



none

The Attempt at a Solution



I'm unsure if this is true because I'm also unsure if it already assumes that it is closed under scalar multiplication and addition. If it is closed, then I'd like to think it is true, but it might be false if it isn't closed. Help me a lil' to know what's already assumed?
Do you not know what the "union" of two sets is? A question like this doesn't "assume" anything it doesn't say. Since a "subspace" is simply a subset that is closed under addition and scalar multiplication, surely a question that asks whether something is a subspace or not will not "assume" that it is a subspace.

As war485 suggested, in R2, {(x, 0)} is a subspace, containing, say, (2, 0). The set {(0, y)} is a subspace containing (0, 3). Their union does NOT contain (2, 3).
 

Related to Linear Algebra: Vector Subspaces

1. What is a vector subspace?

A vector subspace is a subset of a vector space that follows the same rules and operations as the original vector space. In other words, a vector subspace is a smaller space within a larger vector space that still maintains the properties of a vector space.

2. How do you determine if a set of vectors is a subspace?

To determine if a set of vectors is a subspace, it must meet three criteria: closure under addition, closure under scalar multiplication, and contain the zero vector. In other words, if you add two vectors in the set, the result must also be in the set, and if you multiply a vector in the set by a scalar, the result must also be in the set. Additionally, the set must contain the zero vector.

3. Can a subspace have more than one dimension?

Yes, a subspace can have more than one dimension. A subspace can have any number of dimensions as long as it satisfies the criteria of a subspace, such as closure under addition and scalar multiplication.

4. What is the difference between a vector space and a subspace?

A vector space is a collection of vectors that follows specific rules and operations, such as addition and scalar multiplication. A subspace is a subset of a vector space that still maintains these rules and operations. In other words, a subspace is a smaller vector space within a larger vector space.

5. How is linear independence related to vector subspaces?

In order for a set of vectors to form a subspace, they must be linearly independent. This means that none of the vectors in the set can be written as a linear combination of the other vectors in the set. Linear independence is necessary for a set of vectors to form a basis for a subspace.

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