Basis and co-ordinates with respect to a basis.

**katedew987** · December 5th 2010, 06:13 AM

I don't know where to go with this question:

'Let r be a fixed real number. Show that {1, x + r, (x + r)²} is a basis for R[x]2 (that's meant to be a subscript 2) and find the co-ordinates a0 +a1 +a2x² (integers are also meant to be subscript) with respect to this basis.'

Thanks in advance.

**HallsofIvy** · December 5th 2010, 07:21 AM

Well, what do you know? You are clearly expected here to know what a "basis" is- it must span the space and be independent. Any "vector" in $R[x]_2$ , the space of polynomials, with real coefficients, of degree 2 or less, is of the form $a+ bx+ cx^2$ . To show that these functions, 1, x+ r, and $(x+ r)^2$ span the space you must show that whatever a, b, and c are, there exist numbers, $\alpha$ , $\beta$ , and [tex]\gamma[/b] such that $a+ bx+ cx^2= \alpha(1)+ \beta(x+ r)+ \gamma(x+ r)^2$ . To show that they are independent, you must show that the only way you can have $\alpha(1)+ \beta(x+ r)+ \gamma(x+ r)^2= 0$ , for all x, is to have $\alpha= \beta= \gamma= 0$ .

**hatsoff** · December 5th 2010, 07:26 AM

Well basically you just want to multiply out $(x+r)^2$ , and use the result to find $a_0,a_1,a_2$ .

We get $(x+r)^2=x^2+2rx+r^2$ , so $x^2=(x+r)^2-2r(x+r)+r^2$ and $x=(x+r)-r$ .

Therefore $a_0+a_1x+a_2x^2=a_0+a_1(x-r)-a_1r+a_2(x+r)^2-2a_2r(x+r)+a_2r^2$

$=(a_0+a_2r^2)+(a_1-2a_2r)(x-r)+a_2(x+r)^2$

EDIT: Ah, looks like Hallsofivy beat me to it.

Thread: Basis and co-ordinates with respect to a basis.

LinkBack

Thread Tools

Display

Basis and co-ordinates with respect to a basis.

Similar Math Help Forum Discussions

Derive the relationship between Unit Polar Basis and Cartesian Basis

find the matrix for ProjL with respect to the basis

matrix A for T with respect to the basis B.

matrix of inner product on span(s) with respect to a ordered basis

Matrix with respect to the standard basis

Search Tags