Hi, I was working on a problem and I can't figure out what I'm supposed to do.(adsbygoogle = window.adsbygoogle || []).push({});

It reads, find the vector in subspace S that is closest to v; write v as the sum of a vector in S and a vector in S^a; and find the distance from v to S.

S spanned by {(1,3,4)} v = (2,-5,1)

Ok, what I did was I used some equation to find a least squares solution to Ax = b, where b is v and A is S.

So I took S and multiplied it by (1,3,4) and obtained 26.

And I then multiplied v by (1,3,4) and obtained -17

So the equation became:

26x = -17

x = -17/26

So when I multiply what I obtained for x and the vector that spans S together, I should get the point closest to v in/on S right?

Now, to find v as a sum of a vector in S and a vector in S^a, I just did the gram schmidt process.

Its long so instead of writing it out I'll just give you my answer:

(1/156)(414,-474,-252)

This vector is orthogonal to the vector that spans S, the dot product is 0.

So, assuming all of that is correct, I reached the point where I have to tell how far v is from S. I know where they're closest, when the vector in S is multiplied by the constant we obtained for x (-17/26). I have an orthogonal vector to S, but what am I supposed to do now? How do I write v as a sum of vectors in S and S^a , and how do I find the distance?

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# Homework Help: Linear Algebra: Least Squares and vectors

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