Proving Subspaces of Finite-Dimensional Vector Spaces

In summary, a subspace is a subset of a vector space that satisfies all the properties of a vector space, including closure under vector addition and scalar multiplication and containing the zero vector. To prove that a subset is a subspace, it must satisfy all these properties. A subspace can have more than one basis, as long as the basis consists of linearly independent vectors that span the subspace. To prove that a subspace is finite-dimensional, it must have a finite number of vectors that form a basis. A subspace of a finite-dimensional vector space cannot be infinite-dimensional because it is a subset of the vector space and therefore cannot have a higher dimension.
  • #1
hkus10
50
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1) How to show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W<= dimV.

2) How to show that if a subspace of a finite-dimensional vector space V and dim W = dimV, then W = V.

3) How to prove that the subspace of R^3 are{0}, R^3 itself, and any line or plane passing through the origin.

How to approach these three Questions?

Thanks
 
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  • #2
You approach any "proof" by looking at the definitions! What is the definition of "finite dimensional" vector space? What is the definition of "dimension" for such a space?
 

Related to Proving Subspaces of Finite-Dimensional Vector Spaces

1. What is a subspace?

A subspace is a subset of a vector space that satisfies all the properties of a vector space. This means that it must be closed under vector addition and scalar multiplication, and it must contain the zero vector.

2. How do you prove that a subset is a subspace?

In order to prove that a subset is a subspace, you must show that it satisfies all the properties of a vector space. This includes showing that it is closed under vector addition and scalar multiplication, and that it contains the zero vector. If all these properties are met, then the subset is considered a subspace.

3. Can a subspace have more than one basis?

Yes, a subspace can have multiple bases. This is because a basis is defined as a set of linearly independent vectors that span the entire subspace. Different sets of linearly independent vectors can span the same subspace, therefore resulting in multiple bases.

4. How do you prove that a subspace is finite-dimensional?

In order to prove that a subspace is finite-dimensional, you must show that it has a finite number of vectors that form a basis for the subspace. This means that the subspace can be spanned by a finite set of linearly independent vectors.

5. Can a subspace of a finite-dimensional vector space be infinite-dimensional?

No, a subspace of a finite-dimensional vector space cannot be infinite-dimensional. This is because a subspace is a subset of a vector space, and therefore, it cannot have a larger dimension than the vector space it is a part of. If the vector space is finite-dimensional, then the subspace must also be finite-dimensional.

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