Linear Algebra: Basis Homework Help

In summary, a basis in linear algebra is a set of linearly independent vectors that span a vector space. To determine if a set of vectors form a basis, we need to check for linear independence and spanning the vector space. A basis is different from a spanning set in that it is a minimal spanning set, while a spanning set may not be linearly independent. A vector space can have more than one basis, as the dimension of the space is not unique to a specific basis. Bases are used in linear algebra to represent vectors, perform operations, find coordinates, and solve systems of linear equations.
  • #1
phyxius117
4
0

Homework Statement


Problem21.jpg


Homework Equations





The Attempt at a Solution



I don't know how to do this problem.. Help please!
 
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  • #2
Here, maybe this will help you understand projections onto a subspace:
http://www.cliffsnotes.com/study_guide/Projection-onto-a-Subspace.topicArticleId-20807,articleId-20792.html

Do you understand the concept of orthogonality?
 
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Related to Linear Algebra: Basis Homework Help

What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be written as a linear combination of the basis vectors. The number of basis vectors is known as the dimension of the vector space.

How do you determine if a set of vectors form a basis?

To determine if a set of vectors form a basis, we need to check two criteria: linear independence and spanning the vector space. Linear independence means that none of the vectors can be written as a linear combination of the others. Spanning the vector space means that every vector in the space can be written as a linear combination of the given vectors. If both criteria are met, then the set of vectors form a basis.

What is the difference between a basis and a spanning set?

A basis is a set of linearly independent vectors that span a vector space. A spanning set is a set of vectors that span a vector space, but they may not be linearly independent. A basis is a minimal spanning set, meaning that it has the smallest number of vectors needed to span the vector space.

Can a vector space have more than one basis?

Yes, a vector space can have more than one basis. This is because the dimension of a vector space is not unique to a specific basis. There can be multiple sets of linearly independent vectors that span the same vector space, and each set can be considered a basis.

How is a basis used in linear algebra?

Bases are used in linear algebra to represent vectors in a vector space. By writing a vector as a linear combination of the basis vectors, we can perform operations such as addition, subtraction, and scalar multiplication. Bases are also used to find the coordinates of a vector and to solve systems of linear equations.

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