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I know that using the usual addition and scalar multiplication conditions, matrix vector spaces are vector spaces.

However, I'm confused by the span of matrix vector spaces. We'll use 2 x 2 for the time being.

I will represent matrices in the following form:

[entry1 entry2

entry 3 entry4] as [entry1 entry2; entry3 entry4]

Let's say for a certain basis B = {[1 0; 0 0], [0 0; 0 1], [0 1; -1 0], [0 1; 1 0]}, I want to show that it spans M_(2x2).

The solution is [a b; c d] = a [1 0; 0 0] + d [0 0; 0 1] + (b-c)/2 [0 1; -1 0] + (b+c)/2 [0 1; 1 0]

I can see how the right hand side equals [a b; c d], where [a b; c d] represents any matrix in M_(2x2), but I am confused as to how to arrive at the 'coefficients' for the basis of matrices.

I still remember how we do it in the vector (m x 1) form, where I simply let A = matrix with the basis vectors as columns and row reduce it, with the solution [a, b, c....]T. However, I'm having extreme trouble applying this analogy to matrix vector spaces. I'm feeling as if that it is not the right approach here. Could someone guide me please? Thanks :)

I know that using the usual addition and scalar multiplication conditions, matrix vector spaces are vector spaces.

However, I'm confused by the span of matrix vector spaces. We'll use 2 x 2 for the time being.

I will represent matrices in the following form:

[entry1 entry2

entry 3 entry4] as [entry1 entry2; entry3 entry4]

Let's say for a certain basis B = {[1 0; 0 0], [0 0; 0 1], [0 1; -1 0], [0 1; 1 0]}, I want to show that it spans M_(2x2).

I still remember how we do it in the vector (m x 1) form, where I simply let A = matrix with the basis vectors as columns and row reduce it, with the solution [a, b, c....]T. However, I'm having extreme trouble applying this analogy to matrix vector spaces. I'm feeling as if that it is not the right approach here. Could someone guide me please? Thanks :)

After graduation Katie is offered a job in Los Angeles at a salary of $80,000 per year and another job in San Francisco at a salary of $60,000 per year. The CPI for Los Angeles is 120. She decides to accept the job offer from San Francisco because she realizes that this offer represents greater purchasing power. This therefore implies that the CPI for San Francisco is:

(a) greater than 90.

(b) less than 90.

(c) greater than 120

(d) greater than 100.

I know that the CPI provides a measure of the overall prices of goods that a consumer typically buys. I'm just wondering about the calculations when used in salaries though.

Thanks in advance!

I recognize that budget deficits would mean that the government spending is greater than the tax revenue attained. However, I don't get the 'financed by' part. Thanks in advance!

So let f: IR --> IR and g: IR --> IR be functions that are continuous at some c in IR, and let r: IR --> IR^2 be given by r(t) = [f(t) g(t)].

I need to prove that r is continuous at c. Also, somewhere I will need to show lim (t --> c) || r(t) - r(c) || = 0.

Not sure what to do.. thanks in advance!

The question is as above. I know from http://symbolab.com/solver/series-calculator/%5Csu... that it converges. However, I have not learnt Raabe's test and am looking for alternative ways to test for the convergence. Thanks in advance!

So the context:

A 60.0ohm resistor is connected in series with a 30.0microF capacitor and a source whose maximum voltage is 120V, operating at 60.0Hz.

There are parts of the question asking about capacitive reactance and blah, which are fine at this point for me.

The main question is: How will adding an inductor in series with the existing resistor and capacitor affect the current? Explain your answer.

So without specifying an inductance, I am wondering what to do, because I know that If X_L = X_C, then the current will reach a maximum value of 120/60 = 2A. However, depending on what the values of X_L is, the impedance can be high/low which would result in a lower/higher current. So am not sure how to properly answer it.

Thanks in advance!

It's as above. I'm not sure how to go about it, although I know it'll diverge.

Hi there, I'm trying to get to the formula but I'm unable to do so..

I have sound algebra skills. By that I mean I know how to expand and simplify well, but I'm susceptible to careless mistakes, of course. However, I think I am missing something else when trying to get to the above formula.

I'll use gamma as g.

I know that E = gmc^2 and p = gmu.

I tried to square both sides, getting down to E^2 = (mc^2)^2 / (1 - v^2 / c^2) and p^2 = (mv)^2 / (1 - v^2 / c^2).

Then I did E^2 - p^2 and ended up with:

(c^2-v^2) (E^2 - p^2) = c^2 [ (mc^2)^2 - (mv)^2 ]

Not sure how to continue on from here?

Thanks in advance!

>>> def nPrint(message, n):

while n > 0:

print(message)

n -= 1

>>> k = 2

>>> nPrint("A message", k)

A message

A message

>>> k

2

I'm not sure why this gives k as 2? Aren't I updating the value for n already in nPrint (i.e. also for k when invoking nPrint)? Or does this boil down to the reason that what's in a function stays in the function? So k = 2 is defined outside of the function, and even though we're updating k in the function, the value of k stays as it is in the function?

Not too sure about this..

I've seen two different formulae for the work done in an adiabatic process (in addition to looking at the internal energy change)...

There's apparently [ (P2V2-P1V1)/(1-k) ] from https://nz.answers.yahoo.com/question/index?qid=20...

and there's P1V1^k * [ ( V2^(1-k) - V1^(1-k) )/(1-k) ], where k is the ratio of Cp/Cv.

What are the conditions that differentiate between the two formulae?

Thanks in advance!

What are some of the reasons for constant residual contribution to the resistivity in pure metals?

Also, with metals and semiconductors, what affects the temperature dependence of resistivity?

Thanks in advance!

So if I let O, P and Q be any point in space which are not collinear, we can consider a triangle OPQ. Let M1 be the midpoint of OP and M2 be the midpoint of PQ.

I'm supposed to prove that the lines between any two vertices and the midpoints of the opposing sides meet at a point S and divides the lines in a common ratio.

I tried to find the vector OM2 and BM1, where OM2 = OP + a(OQ-OP), where a e IR.

QM1 = OQ - OM1 = OQ - OA/2

Then I tried getting the parametric vector equations for the two lines,

where OM2 = OP + a(OQ-OP) and

L2 = bOQ - OP/2 with b e IR.

Now I want to show that where S is the intersection of L1 and L2, OS = cOM2 and QS = cQM2 and to find the value of c.

I think I've got the set-up wrong, but I'm not sure how to get the right parametric vector equations.

Thanks in advance!

So say we let R and S be any n x n matrix, show that if T = RSR^-1, then we get that det(T - aIn) = det(S - aIn) for all a e IR.

In is the identity matrix, with dimensions n x n.

I wanted to go with T = SRR^-1 and hence T = S, but I know that is not right since RS != SR. So I'm not sure how I'd go about this...

I'm also trying to show from left to right, instead of working from the right.

Thanks in advance!

I think it's meant to be a simple question, but I'm not sure how to carry on.

Basically, I'm given that Cv = 314 J/kg-K. With that, I am to find the mass of the argon atom.

I tried thinking about Eint = 3/2 kT and trying to get rid of T as a variable. Howver, not sure what to compound it with. I thought of Qv = nCv dT, but since Q is a transfer variable and I want T to be constant, I can't use Qv...

Might be something i haven't learnt yet...

thanks in advance!

How would I go about it?

do I start by squaring both sides of the inequality? Or are there some restrictions on when we can square an inequality (on both sides)?

Thanks in advance!

The answer is presumably x = 0 or x > 1, but how do I write the working for it?

I'm not sure how to go about the squaring of both sides of the inequality..

Thanks!