Finding the Complements of W in R^4 to Orthogonal Vectors and Systems

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In summary, the purpose of finding the complements of W in R^4 is to create an orthogonal basis for a vector space, simplifying calculations and representation of vectors. The complements can be found using the Gram-Schmidt process, which involves finding orthogonal vectors that span the same subspace as W. Orthogonal vectors are perpendicular, while orthogonal systems are both perpendicular and normalized. An example of finding complements of W in R^4 is using the Gram-Schmidt process to find two orthogonal vectors, such as (2, -1, 1, 0) and (0, 0, 1, -1), that form a basis for the complement of W. Having an orthogonal basis is important for simpl
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transgalactic
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W is sub space of [tex]R^4[/tex] which is defined as
http://img21.imageshack.us/img21/1849/63042233.th.gif

find the system that defines the complements [tex]W^\perp [/tex] of W
i have solved the given system and i got one vector (-1,1,0,0)
so its complement must be of R^3 and each one of the complements vectors are
orthogonal to it

how to find them
??
 
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solved it :)
 

FAQ: Finding the Complements of W in R^4 to Orthogonal Vectors and Systems

What is the purpose of finding the complements of W in R^4?

The complements of W in R^4 are used to create an orthogonal basis for a vector space. This allows us to easily represent vectors and perform calculations in the space.

How do you find the complements of W in R^4?

To find the complements of W in R^4, you can use the Gram-Schmidt process. This involves finding a set of orthogonal vectors that span the same subspace as W. Then, you can use these vectors to create a basis for the complement of W.

What is the difference between orthogonal vectors and orthogonal systems?

Orthogonal vectors refer to a set of vectors that are perpendicular to each other, while an orthogonal system refers to a set of vectors that are both perpendicular and normalized (unit length). In other words, an orthogonal system is a set of orthogonal vectors that form an orthonormal basis.

Can you provide an example of finding the complements of W in R^4?

Sure, let's say W is the subspace spanned by the vectors (1, 2, 0, 0) and (0, 0, 1, 1). To find the complements, we can use the Gram-Schmidt process to find two orthogonal vectors that span the same space as W. One possible solution is (2, -1, 1, 0) and (0, 0, 1, -1). These two vectors are orthogonal and form a basis for the complement of W.

Why is it important to have an orthogonal basis for a vector space?

An orthogonal basis allows us to easily represent vectors and perform calculations in a vector space. It also simplifies the process of finding the projection of a vector onto a subspace, which has many practical applications in fields such as physics and engineering.

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