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phyxius117
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I don't know how to do this problem.. Help please!
Linear Algebra: Basis Homework Help
- Thread starter phyxius117
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- Algebra Basis Linear Linear algebra
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.
I don't know how to do this problem.. Help please!
- #2
silvermane
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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?
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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FAQ: 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.