Orthogonal Projection Problems?

[ashah99]

Summary:: Hello all, I am hoping for guidance on these linear algebra problems.
For the first one, I'm having issues starting...does the orthogonality principle apply here?
For the second one, is the intent to find v such that v(transpose)u = 0? So, could v = [3, 1, 0](transpose) work?

 

[Office_Shredder]

For the second I think you have a correct answer (there are many choices). For the first, what is the orthogonality principle?

 

[ashah99]

For the second I think you have a correct answer (there are many choices). For the first, what is the orthogonality principle?

The way it was explained to me was that given a vector x in a sub space, find a closest point x_hat that is in S. Not sure if that’s the right approach for the first problem.
And yes I think you’re right on the second part on numerous answers, I wanted confirmation so thanks for that. Another answer I can think of is [4 0 -1]^T.

 

[Office_Shredder]

The way it was explained to me was that given a vector x in a sub space, find a closest point x_hat that is in S. Not sure if that’s the right approach for the first problem.

That certainly sounds like what you're trying to do. Why don't you try it out and post your computation here?

 

[ashah99]

That certainly sounds like what you're trying to do. Why don't you try it out and post your computation here?

Not sure about the formula to use. Have any suggestions?

 

[Office_Shredder]

Not sure about the formula to use. Have any suggestions?

Just try to do this directly and ignore any fancy words. Can you write down an expression for arbitrary vector in the span of 1 and$e^t$? Then try to compute the norm. You should get a quadratic formula in two unknown variables.