Find orthogonal base for u+v without answer

• transgalactic
In summary, the conversation discusses finding the orthogonal base for the sum of two vectors in R4. The vectors v1, v2, v3, and u1 are given and are independent, but not orthogonal. The solution suggests using the standard basis for R4 as it is orthonormal and satisfies the conditions of the problem. However, the speaker questions why the given vectors cannot be used as a basis instead.
transgalactic
i got these vectors R4:
v1=(1,1,0,1)
v2=(0,-4,2,0)
v3=(0,0,-18,0)
u1=(1,1,0,1)
u2=(-1,0,1,0)

in the row reduction process v1 v2 v3 u1 are left independent
i need to find the orthogonal base of u+v?

in the solution i was told that because v1 v2 v3 u1 are independent

then the orthogonal base of u+v the standard base of is
(1,0,0,0)
(0,1,0,0)
(0,0,1,0)
(0,0,0,1)

but why take the standart base
the vectors v1 v2 v3 u1 are independent they can act as a base
??

You have left some things out. Are we to assume that v is spanned by v1, v2, and v3? And that u is spanned by u1 and u2?

Yes, v1, v2, v3, u1 are independent and so are a basis for u+v. But they are not orthogonal. Since there are four of them, they span all of R4. You know that the standard basis for R4 is orthonormal so it satisfies the conditions of the problem: Find an othogonal basis for u+ v.

1. What does "orthogonalized base" mean?

Orthogonalized base refers to a set of vectors that are perpendicular to each other. In other words, they form a right angle with each other. This property is useful in linear algebra and allows for easier calculations and analysis.

2. Why is it important to find an orthogonalized base?

Finding an orthogonalized base is important because it simplifies mathematical operations and makes it easier to understand and analyze vector spaces. It also helps in finding the solutions to systems of equations and can reduce computational errors.

3. How do you find an orthogonalized base?

To find an orthogonalized base, you can use the Gram-Schmidt process, which is a mathematical algorithm that takes a set of linearly independent vectors and produces an orthogonal set of vectors. Another method is to use the QR decomposition, which factorizes a matrix into an orthogonal matrix and an upper triangular matrix.

4. Can any set of vectors be orthogonalized?

No, not all sets of vectors can be orthogonalized. For a set of vectors to be orthogonalized, they must be linearly independent. This means that none of the vectors can be expressed as a combination of the other vectors in the set.

5. What are some applications of orthogonalized bases?

Orthogonalized bases have various applications in mathematics, engineering, and computer science. They are used in signal processing, image compression, and data analysis. They are also used in the formulation of quantum mechanics and in optimization algorithms.

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