Question

see attachment , please, Problem 5 I don't understand why the problem indicated about the matrix when we can get the answer from u,v w only.

Answer 1

when applying the inner product, I get <u,v> = -1 but it's wrong. my book said that it's = -5. so I know my understanding is wrong but don't know why.

Answer 2

complex conjugate of the second vector. but your vectors seem real

Answer 3

it's the part of weighted Euclidean Inner Product.

Answer 4

hmm. "-1" looks right

Answer 5

@Hoa you sure you looked up the right answer?

Answer 6

yes, sir.

Answer 7

\[
u'=[2\;1;1\;1][2;1]=[4+1;2+1]=[5;3]\\
v'=[2\;1;1\;1][-1;1]=[-2+1;-1+1]=[-1;0]\\
<u',v'>=(5)(-1)+(3)(0)=-5
\]
missed the REGION!!!!

Answer 8

first project the vectors on to your region, then perform the operations

Answer 9

would you please show me how to solve problem 4 from the first attachment?

Answer 10

I think what they are asking is to use the vectors from "3" to find the Euclidean inner product.

Answer 11

Thanks a lot. I appreciate what you offer.

Answer 12

ok, so we know that the eucledean inner product is due to projection.
Let the projection be generated by "A"
\[<u',v'>=<(Au),(Av)>\]

Answer 13

i think we can take "A" as a Lie group generator!

Answer 14

I make up the matrix A as \[\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\] and gain Au. Av but I stuck at 3 equations, 4 variables.

Answer 15

my equations are: \[a^2+c^2 =3\]
\[b^2+d^2=5\]
ab + cd =0

Answer 16

the solution sheet gives out the answer is \[\left[\begin{matrix}\sqrt{3} & 0 \\ 0 & \sqrt{5}\end{matrix}\right]\]

Answer 17

I step away awhile to eat something. I sit here from 12 until now. will be back when finish eating.

Answer 18

the matrix corresponding to the inner product <u,v> = w1u1v1 +w2u2v2 +.... is \[\left[\begin{matrix}\sqrt{w1} & 0 \\ 0 & \sqrt{w2\\}\end{matrix}\right]\]

Answer 19

sorry friend.

Answer 20

like this:
\[\left[\begin{matrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{matrix}\right]\]

Answer 21

this is the Lie Group generator.

Answer 22

Group theory has amazing applications. Matrix representations simplify the expression of information and provide new insights. The above Lie Generator causes rotation of a system. similarly, you can have translation, scaling, etc.. and for N-dimensions

Answer 23

haha. please, dont embarrass me by calling "sir". I am just another student of the nature.

Answer 24

lol. how about continue with 'kid'..

Answer 25

btw, help me figure out how to find d(A,B) where A,B are matrices

Answer 26

distance = norm of difference (vectors)

for matrices, you have Frobenius norm, infinite norm, etc..

Answer 27

Frobenius norm =\(\sum\limits_{i,j}(a_{ij}-b_{ij})^2\)

Answer 28

is the common one