Question

Help. See attahchment

Answer 1

I know for letter a it no invertible. I need help with 5b 5c 5d 5e

Answer 2

@pooja195

Answer 3

@pooja195

Answer 4

@518nad

Answer 5

@Nnesha

Answer 6

for b you need to calculate the derative of each basis

Answer 7

1'=0,x'=1, x^2'=2x, x^3'=3x^2

Answer 8

\[\left[\begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 &2& 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 &0 &0\end{matrix}\right]\]

Answer 9

this is you matrix h1

Answer 10

h2 is h1*h1

Answer 11

omg thanks. but what do I do when a question says represent a matrix on the basis...

Answer 12

with question d, do you add the two matric with h_1 and 2*h_2?

Answer 13

@caozeyuan

Answer 14

yes, and for e you need to express the derivative of each basis as a column vector, then the matrix whose column are those columnn vectors is the h1 under this basis

Answer 15

e.g 1 is (1,0,0,0) and 1'=0 which is (0,0,0,0)

Answer 16

so (0,0,0,0) is the left most column of your h1

Answer 17

(1+x)'=1 so your second to the left one is (1,0,0,0)

Answer 18

so after you took the derivatives of the basis they give you in e, write down the coefficents of the derivative as a column vector with the first componenet being contant term and last componet being the coeff for x^3

Answer 19

Ok but how do I know which number to plug in. Going back to question d, I need to add the two representation together or the function?

Answer 20

h2 is just h1*h1, and I've already wirttern h1 as a matrix which you can just do the matrix multiplication to get h2

Answer 21

then you have a matrix which is 2*h1 and you add these two together to get the answer

Answer 22

2h1 is not the same as h1×h1

Answer 23

but h2 is

Answer 24

you also need to add 2*h1 to the h2 you just calculated from h1*h1

Answer 25

Ok I got it. How do represent on question e?

Answer 26

@caozeyuan

Answer 27

hang on a second, eating dinner right now be back in half a hour

Answer 28

!!! I got vector 1000,1100,1110,1111

Answer 29

yes, and now you have to deffientiate each basis to get the basis after the transformation

Answer 30

for example, 1000 becomes 0000 becuase 1'=0

Answer 31

Wait I thought the question ask to just represent

Answer 32

Will I get 0000,0100,0120,0123 ?

Answer 33

Yes you need to represent the transformtion, nott the basis of your original vector space

Answer 34

\[T:V \rightarrow W\]

Answer 35

this linear transformation maps the basis of V to basis of W

Answer 36

your basis in V is given, tey are 1000, 1100, 1110, 1111

Answer 37

and you can obtain basis in W by computing the derivative, they are 0000, 1000, 1200, 1230

Answer 38

so 0 1 1 1
0 0 2 2
0 0 0 3
0 0 0 0

Answer 39

this is your matrix representation of T

Answer 40

notice if you let T act on each basis vector in V it returns the corresponding basis vector in W

Answer 41

But when you take the derivative you would get 0100 in the second column

Answer 42

I just relized your calculation of basis of W is wrong

Answer 43

no, you get 1000

Answer 44

this is your (1+x)' which js just 1, hence your write 1000

Answer 45

What the answer then?

Answer 46

0 1 1 1
0 0 2 2
0 0 0 3
0 0 0 0

Answer 47

Show me how you got that