These are all 2x2 matrices right? If yes, the first basis looks like the basis for the set of all 2x2 matrices with real coeffieicnts
So I decided to check out the answers after all and for the first problem, it said that the transformation matrix is
1 0 0
0 -1 0
0 0 1
Which I’m trying to figure out why.
The second problem is more arithmetic based since you’re multiplying the matrices
[1 1]M
[1 1]
and
M[1 1]
[1 1]
with the each of the four give bases by plugging them in M and then subtracting it, but the answer I keep getting is totally different and I know for fact that the problem is not with my vector multiplication so I just wanted to check with you to see if you would get the same answer as the book, which is…
-2 0 0 0
0 -2 0 0
0 0 0 0
0 0 0 0
Linear Algebra is only hard because it’s so simple. Weird as that seems.
As I mentioned, your first basis is a basis for the basis for the set of all 2x2 matrices with real coefficients. This has dimension 4. P_3 also has dimension 4, so these two vector spaces are isomorphic. You can assign:
1 to your first matrix
x to the second
x^2 to the third
and x^ 3 to the fourth matrix that you presented, in that order.
P_2 is contained in P_3, so if you let the coefficient of x^3 be zero, then so is the coefficient of the corresponding basis vector in M_(2x2). So you can express everything in P_2 in terms of the basis vectors for M_(2x2) in this way.
As for the transformation matrix itself, P_2 is isomorphic to R^3. R^3 has the standard basis vectors <1,0,0> , <0, 1, 0>, and <0,0,1>
now associate
x^2 with <1, 0, 0>
x with <0, 1, 0>
and
1 with <0, 0, 1>
then -x is associated with <0, -1, 0>
Join these together in a column matrix and you have the transformation matrix you requested.
I don’t understand the notation on the transformation matrix for the second problem.
The matrices
1 1
1 1
are being multiplied by M, which is where you plug in the basis matrices one by one. So for the first basis matrix, I would set it up as
[1 1][1 1]
[1 1][-1 -1]
subtracted by
[1 1][1 1]
[-1 -1][1 1]
After you do the arithmetic to see what numbers you would get in the 2 x 2 matrix, that will dictate what you will get for the first column of the transformation matrix. You do the same with the second, third, and fourth given basis matrices to find the second, third, and fourth columns of the transformation matrix respectively.
And just to remind you, the transformation matrix is again
[1 1]M
[1 1]
minus
M[1 1]
[-1 -1]
Sorry if this is confusing.
Just completed math 101 and now I’m taking 50. I is moving up in the earth!!! Haven’t taken any math classes in years so starting over and learning this stuff again is pretty interesting. Math isn’t hard people, you just forgot how to do it. You can pretty much teach yourself with enough time and patience. Taking these classes online has taught me that. Will I forget it all again when I’m done meeting my requirements? Probably.
I usually forget stuff after each semester, but it’s during the first few weeks of a new math class, ie. 150 -> 151, 151 -> 252, etc. is when stuff starts comeing back to you.
I find mathematics interesting but sometimes I feel like I’m reading hieroglyphics.
Did you all always have harmony with math? Or did somewhere down the line you guys really took initiative to conquer math? It suck trying to handle calc, seeing the problem then forgetting everything you know. But I am serious about retaining this info
Math/Science were always my favorite subjects in school so I generally did well. However, I didn’t gain a true appreciation for math until recently. If you really want to retain the information that you’ve learned, you have to look beyond the numbers and computations. Unfortunately, (coming from an undergrad) this is how math is taught in schools, so you’re going to have to put in more of your own time if you really want to study the subject.
It doesn’t really matter what the material is, if you want to learn something, you have to take the initiative and learn it. Math is no exception. If you really want to understand, you have to go beyond thinking “here is a problem and this is it’s solution” to “here is how these are related.”
Math rewards your discipline with new ways to apply what you’ve learned to new areas of math and punishes laziness by rendering you blind to what lies before you.
Props to Newton, Fourier, Pascal, and etc… Those guys were so skilled mathematicians / physicists that they could look at a physical phenomenon, say heat transfer in a rod or vibration in a string, and approach and derive a expression that would represent the said physical phenomenon and then also derive a solution to it.
I hated math so much I ended up liking it. Have to start over since I’m changing my major.
Ummm…yeaah, just want to subscribe.
I was really stoked when I could do factorial ANOVAs.
Although I’ll probably never do another one out by hand since everyone cheats and uses SPSS.
I’ve always enjoyed math and have always been good at it. Easiest subject for me by far.
how does one go about finding the domain of the logarithm
ln x+ln(2-x)
Two keys to remember-
- if a=ln(x), then e^a=x. (If using any other base log, like log10 or log2, replace the ‘e’ with that base number)
- Since e is a positive number, no matter what number, no matter how small, positive, or negative ‘a’ is, e^a will ALWAYS > 0.
Therefore, whatever ‘x’ you use in ‘ln(x)’ MUST be >0, else the log pukes.
For the first half, ‘ln(x)’, x must be >0. So you know the domain has one end at ‘0’.
For the second half, ‘ln(2-x)’, if x=>2, then the log would be less than zero and break. So x<2
X must be >0 for the first half and x<2 for the second half, so the domain is 0<x<2.
By definition, the domain of the ln(x) is all positive numbers,
For ln(2-x) the domain is 0 < 2-x, which implies that x < 2
So for the function f(x) = ln(x) + ln(2-x) the domain (ie the values of x on which this function is defined) is all positive numbers less than 2, or 0 < x < 2.
Fuck what you heard, nerds. Made a chem thread.
Any else check out this channel?
http://www.youtube.com/numberphile
Been watching it here and there and its very entertaining. Basically just videos on number theory/parlor math from what I’ve seen, presented by some brits with Math doctorates