Prove Finite Dimensionality of Subspace of Finite Dimensional Vector Space

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In summary, to prove that the subspace of a finite dimensional vector space is finite dimensional, one cannot assume that the # of basis vectors in the subspace is finite. Instead, one can use the equation dim(V)=#basis vectors and show that the dimension of the subspace is less than or equal to the dimension of the vector space, thus proving that the subspace is finite dimensional. Additionally, it is not possible for a finite-dimensional space to have infinitely many linearly independent vectors.
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torquerotates
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Homework Statement

Prove that the subspace of a finite dimensional vector space is finite dimensional

Homework Equations

dim(V)=#basis vectors

The Attempt at a Solution


I was thinking about making a summation of the basis vectors in the vector space and the subtracting the summation of the basis vectors in the subspace. That way I get a finite# of basis vectors left over. But that thinking turned out to be flawed because I'm assuming that the #of basis vectors in the subspace is finite(what I'm trying to prove). So I'm back to square 1.
 
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Can a finite-dimensional space have infinitely many linearly independent vectors?
 

Related to Prove Finite Dimensionality of Subspace of Finite Dimensional Vector Space

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 is closed under vector addition and scalar multiplication, and contains the zero vector.

What does it mean for a subspace to be finite dimensional?

A subspace is finite dimensional if it has a finite basis. This means that there is a finite set of vectors that can be used to span the entire subspace.

How do you prove the finite dimensionality of a subspace?

To prove the finite dimensionality of a subspace, you can show that it has a finite basis. This can be done by showing that any vector in the subspace can be written as a linear combination of a finite set of vectors. Alternatively, you can show that the subspace has a finite dimension, meaning that the maximum number of linearly independent vectors is finite.

Can a subspace of a finite dimensional vector space be infinite dimensional?

No, a subspace of a finite dimensional vector space can never be infinite dimensional. This is because if a vector space is finite dimensional, then all its subspaces must also be finite dimensional.

Why is the finite dimensionality of a subspace important?

The finite dimensionality of a subspace is important because it tells us about the structure and properties of the subspace. It also allows us to use techniques and theorems specific to finite dimensional vector spaces, making it easier to solve problems and prove results.

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