Linear Algebra Proof of Span and orthogonal vector space

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Start date Apr 13, 2012
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Algebra Linear Linear algebra Orthogonal Proof Space Span Vector Vector space

In summary, the span of a set of vectors in linear algebra is the set of all possible linear combinations of those vectors, which can be found by multiplying each vector by a scalar and adding them together. To prove that a set of vectors spans a vector space, it must be shown that every vector in the vector space can be written as a linear combination of the set of vectors. An orthogonal vector space is a vector space where all the vectors are perpendicular to each other, meaning that the dot product of any two vectors in the space is equal to zero. To prove that a set of vectors is orthogonal, it must be shown that the dot product of any two vectors in the set is equal to zero. It is not possible for a set

Apr 13, 2012

sbj5533

Let w=span(w1, w2, ...,wk) where wi are vectors in R^n. Let dot product be inner product for R^n here. Prove that if v*wi=0 for all i-1,2,...,k then v is an element of w^upside down T (w orthogonal).

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Apr 13, 2012

micromass

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Take an element x of W. You must prove that v.x=0.

Try to prove this by expressing x as a linear combination of the w_j.

Related to Linear Algebra Proof of Span and orthogonal vector space

1. What is the definition of span in linear algebra?

The span of a set of vectors in linear algebra is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be created by multiplying each vector by a scalar and adding them together.

2. How do you prove that a set of vectors spans a vector space?

To prove that a set of vectors spans a vector space, you must show that every vector in the vector space can be written as a linear combination of the set of vectors. This can be done by setting up a system of equations and solving for the coefficients of the linear combination.

3. What is an orthogonal vector space?

An orthogonal vector space is a vector space where all the vectors are perpendicular to each other. This means that the dot product of any two vectors in the space is equal to zero.

4. How do you prove that a set of vectors is orthogonal?

To prove that a set of vectors is orthogonal, you must show that the dot product of any two vectors in the set is equal to zero. This can be done by calculating the dot product and showing that it equals zero.

5. Can a set of non-orthogonal vectors span an orthogonal vector space?

No, a set of non-orthogonal vectors cannot span an orthogonal vector space. In order for a set of vectors to span an orthogonal vector space, they must be orthogonal to each other. If the set of vectors is not orthogonal, then the span of those vectors will not be an orthogonal vector space.