Find a Basis for R^4 Subspace Spanning S

Thread starter xsqueetzzz
Start date Nov 22, 2010
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Basis

In summary, a subspace is a subset of a vector space that follows all the properties of a vector space. This includes closure under vector addition and scalar multiplication, as well as containing the zero vector. The dimension of a subspace is the number of vectors in a basis for that subspace, which is also equal to the number of linearly independent vectors in the subspace. A subspace spans a vector space if all the vectors in the vector space can be written as a linear combination of the vectors in the subspace. To find a basis for a subspace, one can use the process of elimination to determine a set of linearly independent vectors. A subspace can have multiple bases, but all bases will have the same number

Nov 22, 2010

xsqueetzzz

Find a basis for the subspace of R^4 spanned by, S={(6,-3,6,340, (3,-2,3,19), (8,3,-9,6), (-2,0,6,-5)

Not too sure where to start.

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Nov 23, 2010

Fredrik

Staff Emeritus

Science Advisor

Gold Member

Start by figuring out if they're linearly independent. If they are, the subspace is 4-dimensional, and those vectors are a basis. If they're not, you can eliminate one and repeat the procedure with the ones you have left.

FAQ: Find a Basis for R^4 Subspace Spanning S

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 is the dimension of a subspace?

The dimension of a subspace is the number of vectors in a basis for that subspace. It is also equal to the number of linearly independent vectors in the subspace.

What does it mean for a subspace to span a vector space?

A subspace spans a vector space if every vector in that vector space can be written as a linear combination of the vectors in the subspace. In other words, the subspace contains enough vectors to reach every point in the vector space.

How do you find a basis for a subspace?

To find a basis for a subspace, you can use the process of elimination. Start with a set of vectors in the subspace and check if they are linearly independent. If not, remove one and check again until you have a set of linearly independent vectors. This set of vectors will be the basis for the subspace.

Can a subspace have more than one basis?

Yes, a subspace can have multiple bases. This is because there are often many different sets of vectors that can span a subspace. However, all bases for a subspace will have the same number of vectors, which is equal to the dimension of the subspace.