Question

Guys how do I find change of basis matrix?

Answer 1

Suppose I have two vector spaces:
B = {1,x,x^2,x^3}
B' = {1, x+1,x^2-2,x^3-x^2}

Answer 2

I meant i have two bases.. damn lin algebra is confusing..

Answer 3

and I just made those two up..

Answer 4

In cases where you are going from the standard basis to another basis, the columns just consist of the elements in the new basis.

Answer 5

So in this case would it be:
1 1 -2 0
0 1 0 0
0 0 1 -1
0 0 0 1

Answer 6

yes

Answer 7

oh okay thanks! What if it was not from standard basis?

Answer 8

You would have to solve some systems.

Answer 9

so it's like in first case to prove that B' was the basis I had to rewrite B as a combination of B'. Now to find change of basis matrix I have to rewrite B' as B?

Answer 10

that is if I'm going from standard to B'

Answer 11

If you think about how a matrix transforms a column vector, the entries in the column vector represent the coefficients for the linear combination of the columns in the matrix.

Answer 12

In this case the columns are the basis vectors.

Answer 13

Each column vector representing elements of the standard basis will be transformed into elements in B'

[1 0 0 0] to [1 0 0 0 ]
[0 1 0 0] to [1 1 0 0 ]
[0 0 1 0] to [-2 0 1 0]
etc

Answer 14

wait do i write them as rows or columns?

Answer 15

columns

Answer 16

\[ \left[ \begin {array}{cccc} 1&1&-2&0\\ 0&1&0&0
\\ 0&0&1&-1\\ 0&0&0&1\end {array}
\right] \]

Answer 17

Just out of curiosity did you learn about row reduction yet in your class?

Answer 18

Yes that's the first thing we learn

Answer 19

Is this right?

Answer 20

I'm not sure you would be required to show that its bijection from the vector space to F^n but I guess that's fine.

Answer 21

we didn't learn bijections and I also have no idea what F^n is.. im pretty sure we never learned that..

Answer 22

I really think they should be teaching the theory in class first. After that everything else would be less confusing.

Answer 23

this is a summer class professor is flying through i just remember how to do row ops, finding inverses and rank and stuff like that.. as soon as the abstract stuff started i got lost.

Answer 24

When I took a linear algebra class the very first thing discussed was the theoretical basis for the subject.

The axioms of a field, of a vector space. Proving all bases for a vector space have the same cardinality to show that the dimension of a VS is well defined. Discussing linear transformations, proving that for T:V->W dim(V) = rank(T) + nullity(T). What properties do one to one and onto transformations have etc. And how a matrix is really just a linear transformation written with respect to bases etc.

Answer 25

we started with matrices and row operations. The first thing we did was solving lin eqns with row ops.. anyway i might have some more questions.. im working on one right now.. these proofs are getting to me..

Answer 26

I wish i had taken a linear algebra course like yours alchemista. mine was like Bahroms. i ended up buying two linear algebra books and reading on my own to try and understand more theoretical concepts. Im a little curious, where did/are you going to school Alchemista?

Answer 27

University of Toronto, it's a Canadian school so you may not have heard of it.

Answer 28

if it's like University of ______ (name of a city or something) i think it must be good.. I just go to a City University of NY.. like a public school in NY - Baruch college

Answer 29

at the school i go to, about 85%-90% of people in the math dept are education majors, so the classes never get too theoretical. which sucks for me =/

Answer 30

Any of u close to graduation yet? I wonder what do u do once u get a math degree.. I dont wanna work in a math field.. Im just minorin in programmin for now..

Answer 31

I should've minored in math.. I would've been almost done takin math classes..

Answer 32

computer science?

Answer 33

i have about a year and a half to go. i just want to do research.

Answer 34

so do I actually.. I mean im tryin to graduate in 3 years which is why im ruining my vacations..

Answer 35

alchemista, something like that.. i guess u could call that computer science..

Answer 36

well I guess it depends, computer science is very different from software engineering (programming)

Answer 37

yea i think im closer to software engineering.. I have to take 3 object oriented programming classes..

Answer 38

java, c++ or python?

Answer 39

C++ I guess. I didn't take any of those yet, but I will start C++ in a week and i don't know what those 3 classes are going to be in. Probably java or C++

Answer 40

If you want to do something fun/interesting you should try taking a course on a functional programming language.

Answer 41

Haskell, scheme, etc.

Answer 42

i dont think ima get a chance to.. my school is a really good business/finance school, but not into math and sciences.. I dont even know why i went there..