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I am studying the book Linear Algebra from Hoffman and Kunze. The authors make the following comment on page 281.

Although it is of limited practical use for computations, it is interesting to note that the Gram-Schmidt process may also be used to test for linear dependence.

I have two question on that comment:

1) Why should I study the Gram-Schmidt orthogonalization process?

2) Is there an example where the Gram-Schmidt orthogonalization process makes easier to prove that a set of vectors is linearly dependent instead of use another method? I've never proved that a subset of vectors was linearly independent by using the Gram-Schmidt orthogonalization process.

Maybe I did not understand what they are saying.

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I think it is indeed unlikely you will really use Gram-Schmidt in all it s gory detail concretely as you move on. My guess is most courses teach it to make sure you understand how the process of finding an orthonormal basis works. To that end it will be useful to actually go through the process by hand a couple of times. Later you will better understand what the software package does in case you go into applications, or you will use the process abstractly in arguments and do it with more confidence ! Bit of a pain but still worth it ! –  Beltrame Feb 12 '12 at 22:12
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The "it" that is "of limited practical use" is not the Gram-Schmidt process as such, only the idea of using Gram-Schmidt to test for linear dependence. –  Henning Makholm Feb 12 '12 at 22:48
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I suppose the point being made by the authors is that if you are in an $n$-dimensional inner product space, and you apply Gram-Schmidt to a set of $n$ vectors which is not linearly independent to start with, you will inevitably obtain the zero vector at some stage in the Gram-Schmidt process, instead of returning an othonormal basis. –  Geoff Robinson Feb 12 '12 at 23:17
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The Gram-Schmidt orthogonalization process is a great thing to learn about because the idea behind it shows up again and again. There are many practical algorithms that use what is essentially a Gram-Schmidt procedure. For example, suppose you have a set $\{v_1, \dots, v_n\}$ of linearly independent vectors (or functions), and you want to approximate another vector (or function) $w$ as a linear combination of the vectors in your set. This can be done by linear regression -- i.e. project $w$ onto the space spanned by the set. An alternative method, which is useful when you have a huge collection of functions in your set, is called "matching pursuit." Here you project $w$ onto the space spanned by the vector $v_i$ that is most correlated with $w$, subtract off that component, then project what remains onto the next most correlated vector from your set, subtract, etc. This process of projecting-subtracting, projecting-subtracting,... is just like Gram-Schmidt, and it is the basic principal behind many practical algorithms for time-frequency decompositions, time-scale (wavelet) decompositions, etc. In other words, there are many "Gram-Schmidt-like" procedures, so you would do well to learn the original!

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The main thing I have seen Gram-Schmidt used for it just to guarantee existence of an orthonormal basis. This makes the basis very nice.

Look through the chapter on inner product spaces, almost all the proofs start off with, let $\alpha_1, ..., \alpha_n$ be an orthonormal basis. This makes the basis really nice because if $(|)$ is your inner product. $$ (\alpha_i| \alpha_j) = \delta_{ij} = \begin{cases} 0 & \text{ if } i \neq j\\ 1 & \text{ if } i = j \end{cases}$$

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