Question

Consider the operation on p3 that takes ax^3+bx^2+cx+d to cx^3-bx^2-ax+d . Does it correspond to a linear transformation from R^3 to R^3 ? If so, what is its matrix?

Answer 1

Is this one of them?

Answer 2

yes sir

Answer 3

by the way abtrehearn i think i might need you to check out the 2nd and third part of my problem the other person was helping me with to see how well its uhh answered after you're all done here if possible XD

Answer 4

I think I mentioned this before. P3 is a 4 dimensional vector space.

Answer 5

It does correspond to a 3 dimensional linear transformation.

Answer 6

* does not

Answer 7

The answer is no, and no matrix exists.

Answer 8

If it was from R^4 to R^4 it would be expressed as follows. Lets have a look:
T(1, 0, 0, 0) =(0, 0, 1, 0)
T(0, 1, 0 ,0) =(0, 1, 0, 0)
T(0, 0, 1, 0) =(1,0 ,0, 0)
T(0, 0, 0, 1) =(0, 0, 0, 1)

Answer 9

Will do.
We seek transformation T that maps
[a, b, c, d] to [c, -b, -a, d]. This is a mapping\[T:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4},\]and its matrix is
[ [0, 0, 1, 0],
[ 0,-1, 0, 0],
[-1, 0, 0, 0],
[ 0, 0, 0, 1]].

Answer 10

The answer, is no since it is asking if there exists a transformation from R^3->R^3

Answer 11

oops forgot the signs.

Answer 12

This is the full question that he posted before:
Consider the operation on p2 that takes ax^2+bx+c to cx^2+bx+a . Does it correspond to a linear transformation from R^3 to R^3 ? If so, what is its matrix? (b) Consider the operation on p3 that takesax^3+bx^2+cx+d to cx^3-bx^2-ax+d . Does it correspond to a linear transformation from R^3 to R^3 ? If so, what is its matrix?

Clearly they are looking for "no" as the answer.

Answer 13

Since the last component d maps to itself, we can look at the subset of polynomials \[B = \left\{ ax^{3} + bx^{2} + cx\right\},\]where a, b, and c are real scalars. Then T:[a,b,c,0] =>[c, -b, -a, 0], and if we lop off the final components, we can have transformation \[U : [a, b, c] \rightarrow [c, -b,-a],\] and its transformation matrix is
[[ 0, 0, 1],
[ 0, -1, 0],
[-1, 0, 0]].

Answer 14

But the original transformation is \[T:\mathbb{R}^4\to\mathbb{R}^4\] I think its a stretch to claim that it corresponds to a \[T:\mathbb{R}^3\to\mathbb{R}^3\] transformation since the image doesn't span the \[\mathbb{R}^4\] vector space and the original transformation does.

Answer 15

You are cutting off part of the image, even if you think that its trivial for the transformation to map a component to itself.

Answer 16

I guess you can claim that corresponds is fuzzy enough to accommodate your interpretation of the problem.

Answer 17

By that I mean the word "corresponds"

Answer 18

In order for U to be a linear fransformation,
\[U([a_1, b_1, c_1] + [a_2, b_2, c_2]) = U([a_1, b_1, c_1] + U(a_2, b_2, c_2])\] and\[U(\alpha[a_1, b_1, c_1]) =\alpha U[a_1, b_1, c_1]).\]for all scalars\[a_1, a_2, b_1, b_2, c_1, c_2, \alpha.\]That's a straightforward procedure. I think you can handle that. Lte me know if you get stuck, though.