How Can Gram-Schmidt Prove dim(w)=dim(V)-1?

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In summary, the conversation discusses proving that the subspace consisting of all vectors v such that <v, x>= 0 (where x is non-zero) has dimension dim(V)- 1 using the Gram-Schmidt process. The process involves selecting a basis for V containing x and performing the Gram-Schmidt process, resulting in a basis containing one basis vector in the same direction as x.
  • #1
theFuture
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So I'm to show that the non-zero vector w={v e V|<v,x> = 0} for all x in V that dim(w)=dim(V)-1.

It recommends using the Gram-Schmidt process to prove this but I tried to work it out and I couldn't make any sense of it. Any suggestions on how to start this out?

[edit]: nevermind, I got it. If you were curious, start by saying x is an element of the set S that is linearly independent and spans V. Then do G-S on V and you find that you lose an element of the set, so there you have it.
 
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  • #2
I think what you mean to say is "prove that the subspace consisting of all vectors v such that <v, x>= 0 (where x is non-zero) has dimension dim(V)- 1."
Select a basis for V containing x and perform a "Gram-Schmidt" starting the process with x as the first vector (so Gram-Schmidt will give a basis containing one basis vector in the same direction as x).
 
  • #3
That's definitely what I meant, but I copied it verbatim from my crappy book (Messer's Linear Algebra) starting with "show the vector . . ." It's good with proofs, not so good with the math.
 

FAQ: How Can Gram-Schmidt Prove dim(w)=dim(V)-1?

What is Gram Schmidt Orthonormalization?

Gram Schmidt Orthonormalization is a mathematical process used to transform a set of linearly independent vectors into a set of orthonormal vectors. This means that the resulting vectors are not only perpendicular to each other, but also have a magnitude of 1.

Why is Gram Schmidt Orthonormalization important?

Gram Schmidt Orthonormalization is important because it allows us to simplify calculations and make them more efficient. Orthonormal vectors have properties that make them easier to work with and can lead to more accurate and efficient solutions in various applications.

What are the steps involved in Gram Schmidt Orthonormalization?

The steps involved in Gram Schmidt Orthonormalization are:

  1. Choose a set of linearly independent vectors.
  2. Normalize the first vector by dividing it by its magnitude.
  3. Find the projection of the second vector onto the first vector and subtract it from the original second vector.
  4. Normalize the resulting vector.
  5. Repeat the process for the remaining vectors, using the previous orthonormal vectors as a basis for the projection.

What are the applications of Gram Schmidt Orthonormalization?

Gram Schmidt Orthonormalization has applications in various fields, such as signal processing, data compression, and numerical analysis. It is also used in solving systems of linear equations and finding the best fit for a set of data points.

Can Gram Schmidt Orthonormalization be applied to any set of vectors?

Gram Schmidt Orthonormalization can be applied to any set of linearly independent vectors. However, it is important to note that there may be cases where the resulting orthonormal vectors are not as accurate or useful due to factors such as numerical instability or the nature of the original vectors.

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