Midterm #6
For midterm question #6, how do we show that {x+1,x-2,x^2+x} is linear and therefore a basis (if it spans the set of polynomials)? I know that one method is to put the set into a matrix form and row reduce, but how can this be done for this set? I also tried using the textbook definition, that $c_{1}v_{1}, c_{2}v_{2},..c_{n}v_{n}=0$ only if $c_{1},c_{2},...,c_{n}=0$ for linearly independent sets. Does this work? Has anyone else tried this method?
Sam_Lai: May 4, 2015, 7:50 p.m.
The textbook definition should work. $c_1(x+1) + c_2(x-2) + c_3(x^2+x) = 0$ iff $c_1,c_2,c_3 = 0$. You can then demonstrate that it spans the set of polynomials with degree less than or equal to two by showing that you can write any polynomial with degree less than or equal to two as a linear combination of those three elements of the set.
Cullen_Im: May 5, 2015, 5:14 a.m.
The bases span $ax^2$ + bx + c when gamma = a, beta = (b-a-c)/3, and alpha = (2b-2a+c)/3. I did this by setting $ax^2$ + bx +c = alpha(x + 1) + beta(x - 2) + gamma($x^2$ +x) and solving for gamma, beta, and alpha.
abbyto1607: May 5, 2015, 3:41 p.m.
I got the same answer. Except, addition that gamma, beta, and alpha will be defined for all a, b, c.