Random math questions

[#1procrastinator]

Assignment question:

Using Cauchy's Mean Value Theorem, prove that for some

Don't know where to start with this one...right-hand side looks like the first few terms of the Taylor series but I don't think that's relevant here 

[kamil9876]

This actually is one form of a Taylor Series error term for a linear approximation. There are general estimates like this for an nth order approximation.

I know that probably doesn't help (or maybe it does?).

[#1procrastinator]

Nope!
We haven't covered Taylor series yet. I think the only way to do for now is to actually use the hint which suggests to consider F(x)=f(x) - f(c) - f'(c)(x-c) and G(x) = (x-c)^2, but I wanted to see if it could be done (and how) without that.

[kamil9876]

I see, that  makes it easier I guess. Without the hint it would probably be a bit of a stretch to come up with it yourself.

[Deleted User]

How do I do this question?
Find the orthogonal projection of (x,y,z) onto the subspace of R^3 spanned by the vectors (1,2,2),(-2,2-1).

I got (5x-2y+4z,-2x+8y+2z,4x+2y+5z), but in the answers there's a 1/9 multiplied to my answer. Maybe I used the wrong formula?

[kamil9876]

I havn't done the calculation myself, but I notice that the two vectors are of norm . So perhaps you forgot to divide out by the norm.

Seeing that you actually GOT an answer (and presumably not the right one) you should actually show us WHAT YOU DID so that we know whether you "used the wrong formula".

[#1procrastinator]

Could I please get some suggestions on how to carry out this integration :p

[Deleted User]

I havn't done the calculation myself, but I notice that the two vectors are of norm . So perhaps you forgot to divide out by the norm.

Seeing that you actually GOT an answer (and presumably not the right one) you should actually show us WHAT YOU DID so that we know whether you "used the wrong formula".

I don't know how to write maths on my computer.

Is this the right formula? projw(u) = <v,u1>u1 + <v,u2>u2 + ... + <v,uk>uk where {u1,u2,...,u3} is an orthonormal basis for W.

[kamil9876]

Yes that is correct, so did you find an orthonormal basis for your subspace? Note that the vectors you started with aren't orthonormal so you can't just take those as your u1,u2 (if that is what you did).

[#1procrastinator]

1) Let A be a 50x49 matrix and B be a 49x50 matrix.  Show that the matrix AB is not invertible.

If A is a 50x50 matrix then shouldn't it be invertible?

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2) If A is an nxn matrix such that , show that is invertible and find an expression for [itex](A+I_n)^{-1}[/itex]

It was suggested I use the geometric series but I haven't learnt that yet so I'm hoping there's an alternative method.

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3) Define [itex]e^A=I_n+A+\frac{1}{2!}A^2+\frac{1}{3!}A^3+...+\frac{1}{2012!}A2012[/itex]
where A is an nxn matrix such that [itex]A^{2013}=0[/itex]. Show that [itex]e^A[/itex] is invertible and find an expression for [itex](e^A)^{-1}[/itex] in terms of A.

...'bout to attempt this one again but I just know I'm not gonna get far...so here it is...

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3) Show that and use this to show that the limit as approaches infinity is infinity. n is any positive integer.

For the first part, I did used the previous result and inserted an 'n' with no justification of where I put it and then multiplied by . Is this valid?
I'm not sure how to do the limit part though.

[kamil9876]

1) This should be a good lesson in thinking of matrices as linear transformations on a vector space rather than some silly box of with numbers! Here is one argument you may use: where Rank is the dimension of the image. But of course the dimension of the image must be 50 dimensional if the matrix was invertible, so it cannot be.

2)

It was suggested I use the geometric series but I haven't learnt that yet so I'm hoping there's an alternative method.

Would proving the geometric series identity without using the phrase "geometric series" count as an "alternative" method? Geometric series is just polynomial factorization really. The polynomial has as a root, hence it must be divisible , therefore there must be some some identity and so . Compute P(X) by equating coefficients and out comes your formula for the geometric series!

3)

What's itex? Btw is this the same from before? Because there is in fact a matrix exponential e^A and it agrees with your formula if . For real numbers x you already know that so it's good to sometimes use such an Ansatz.

[Deleted User]

How do I prove that this transformation is linear? T:R^3 -> M2,2 given by T(x,y,z)=[y z; -x 0]?

And by [y z; -x 0] I mean
[y  z]
[-x 0]

[humph]

1) Let A be a 50x49 matrix and B be a 49x50 matrix.  Show that the matrix AB is not invertible.

If A is a 50x50 matrix then shouldn't it be invertible?

------

2) If A is an nxn matrix such that , show that is invertible and find an expression for [itex](A+I_n)^{-1}[/itex]

It was suggested I use the geometric series but I haven't learnt that yet so I'm hoping there's an alternative method.

------

3) Define [itex]e^A=I_n+A+\frac{1}{2!}A^2+\frac{1}{3!}A^3+...+\frac{1}{2012!}A2012[/itex]
where A is an nxn matrix such that [itex]A^{2013}=0[/itex]. Show that [itex]e^A[/itex] is invertible and find an expression for [itex](e^A)^{-1}[/itex] in terms of A.

...'bout to attempt this one again but I just know I'm not gonna get far...so here it is...

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3) Show that and use this to show that the limit as approaches infinity is infinity. n is any positive integer.

For the first part, I did used the previous result and inserted an 'n' with no justification of where I put it and then multiplied by . Is this valid?
I'm not sure how to do the limit part though.

Bahahaha. Are these MATH1115 questions? I remember teaching these two years ago (the only difference was that 2013 was replaced by 2011, for obvious reasons...).

For what it's worth, most people got these wrong (though that may have been my fault for talking about exponentials and logarithms - people tried to think about those without realising that they are quite different notions when defined for matrices).

[Deleted User]

What are the eigenvalues for this matrix?
2 -3 6
0 5 -6
0 1 0

I got 2,2,3,5 but the answer only accepts 2,2,3 and not 5.

[b^3]

(expanded down the first column)
Not sure where you got the from.