Linear Algebra: Proving Basis of P2 and R3 Through Scalar Multipliers

[stunner5000pt]

Show taht ${a + bx + cx^2, a_{1} + b_{1} x + c_{1} x^2, a_{2} + b_{2} x + c_{2} x^2}$ is a basis of P2 iff {(a,b,c) , (a1,b1,c1) , (a2,b2,c2)} is a basis of R3

suppose ${a + bx + cx^2, a_{1} + b_{1} x + c_{1} x^2, a_{2} + b_{2} x + c_{2} x^2}$ is a basis of P2 then
a linear combination of those three vectors would require all the scalar multipliers to be zeros

but I am not sure where to go from there... do i have to somehow write he basis of P2 as a basis of R3??

DOes the same apply for the only if part?

Determien whether the transformation has an iverse and if so then find the action of its inverse

T; R4 - > R4
T(x,y,z,t) = (x+y,y+z,z+t,t+x)
Both the preimage and the image have the same dimension i have to show that either t is onto or one to one

how owuld i show it is onto? Or one to one?? Do i simply line up x1s and y1s and see if they are equal if the image of them is equal??

 

[0rthodontist]

For the first one, you just have to express yourself more clearly. You want to show that the P2 vectors are independent iff the R3 vectors are independent. If the P2 vectors (call them v1, v2, v3) are independent, then there is no combination av1 + bv2 + cv3 = 0 unless a, b, c = 0. How do you translate that into R3?

For the second one a good idea is to find the matrix of the transformation and see if it's invertible.

 

[stunner5000pt]

ok for the first one... so since those three vectors are independent, then v1,v2,v3 form a basis for R3 don't they?


how do i find the matrix of a transform?? I m not use how to do this since it is not givne in the text

 

[0rthodontist]

No, if v1 v2 and v3 as I defined them are independent, they form a basis for P2. What you need to show (half of what you need to show) is that if v1, v2, and v3 are independent, then the corresponding vectors in R3 are independent.

To find the matrix of T, write (x, y, z, t) as a column vector to the right of the matrix. You know that T is from R4 to R4 so what will be the dimensions of the matrix? Now, you know that the first coordinate of the product will be x + y. Think about how the first coordinate in the product is formed. How can you adjust the matrix so that the first coordinate is x + y? Here's a hint: every entry in the matrix will be either a 0 or a 1.