matrix of a linear map

[bluemath]

Hello,

Linear map f R2 [x] ---> R₂ [x]
f(ax² + bx + c) = (-3a + b)x² + (a - 2c)x + (a- b + c)

I would like to find the matrix of this linear map.

I think it's :

-3a + b	0	0
0	a - 2c	0
0	0	a - b + c

Is that correct ?

 

[Ishuda]

Hello,

Linear map f R2 [x] ---> R₂ [x]
f(ax² + bx + c) = (-3a + b)x² + (a - 2c)x + (a- b + c)

I would like to find the matrix of this linear map.

I think it's :

-3a + b	0	0
0	a - 2c	0
0	0	a - b + c

Is that correct ?

Given that matrix, you certainly have
[x², x, 1] A = [(-3a + b) x², (a - 2c) x, (a- b + c)]
which has the equivalent polynomial p(x) = (-3a + b)x² + (a - 2c)x + (a- b + c). But what is f(x)? Assuming it is infinitely differentiable around zero, i.e. can be represented by a polynomial around zero, and it were a quadratic (or higher), then f(a x² + b x + c) would be of fourth degree (or higher). Since the above equation is only a second degree equation, f(x) must be linear or a constant. Thus
f(ax²+bx+c) = \(\displaystyle \alpha\) (ax²+bx+c)+\(\displaystyle \beta\) = (-3a+b)x² + (a-2c)x + (a-b+c)

Given all that, one (at least this one) comes to the conclusion that A does (partially) represent a linear map if f(x) is a polynomial of degree 1 or less.

 

[pka]

Linear map f R2 [x] ---> R₂ [x] , f(ax² + bx + c) = (-3a + b)x² + (a - 2c)x + (a- b + c)
I would like to find the matrix of this linear map.

-3a + b	0	0
0	a - 2c	0
0	0	a - b + c

Is that correct ?

I admit that there is no absolute standard notation.
I think it is usual to see the basis as \(\displaystyle \{1,~x,~x^2\}\) this
\(\displaystyle \left[ {\begin{array}{*{20}{r}} 1&{ - 1}&1 \\ 1&0&{ - 2} \\ { - 3}&1&0 \end{array}} \right]\)\(\displaystyle \cdot\left[ \begin{array}{*{20}{c}} a \\ b \\ c \end{array} \right]\)

 

[bluemath]

Thanks for your 2 answers.

Pka :

I think not about your matrix because we have no more f R₂[x] ---> R₂[x] (in your last vector, your change 1,x,x² by a,b,c)

 

[pka]

@Pka :
I think not about your matrix because we have no more f R₂[x] ---> R₂[x] (in your last vector, your change 1,x,x² by a,b,c)

I have no idea what you mean by that.
Let \(\displaystyle F = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&1 \\ 1&0&{ - 2} \\ { - 3}&1&0 \end{array}} \right]\)

\(\displaystyle f(5x^2+x-2)\) is \(\displaystyle F \cdot \left[ {\begin{array}{*{20}{r}} 5 \\ 1 \\ { - 2} \end{array}} \right] = \left[ {\begin{array}{*{20}{r}} 2 \\ 9 \\ { - 14} \end{array}} \right]\) or \(\displaystyle -14x^2+9x+2\)

I began by telling you that there is no standard notation. This is the idea of what the transform matrix looks like-not what you posted. BUT in any case, you must follow your notes or textbook.

 

[bluemath]

I understood !

Many thanks for your help, that's great !