Linear Algebra, Theory And Applications, 2012a

224 VECTOR SPACES AND FIELDS
where the polynomials q i (x) are relatively prime and all the polynomials p (x) and
q i (x) have coefficients in a field of scalars F. Thus there exist polynomials a i (x)
having coefficients in F such that
m∑
1= a i (x) q i (x)
Explain why
i=1
R (x) = p (x) ∑ m
i=1 a i (x) q i (x)
=
q 1 (x) ···q m (x)
m∑
i=1
a i (x) p (x)
∏
j≠i q j (x)
Now continue doing this on each term in the above sum till finally you obtain an
expression of the form
m∑ b i (x)
q i (x)
i=1
Using the Euclidean algorithm for polynomials, explain why the above is of the form
M (x)+
m∑
i=1
r i (x)
q i (x)
where the degree of each r i (x) is less than the degree of q i (x) andM (x) is a polynomial.
Now argue that M (x) =0. From this explain why the usual partial fractions
expansion of calculus must be true. You can use the fact that every polynomial having
real coefficients factors into a product of irreducible quadratic polynomials and linear
polynomials having real coefficients. This follows from the fundamental theorem of
algebra in the appendix.
43. Suppose {f 1 , ··· ,f n } is an independent set of smooth functions defined on some interval
(a, b). Now let A be an invertible n × n matrix. Define new functions {g 1 , ··· ,g n }
as follows.
⎛
g 1
⎞ ⎛
f 1
⎞
⎜
⎝
.
g n
⎟ ⎜
⎠ = A ⎝
.
f n
⎟
⎠
Isitthecasethat{g 1 , ··· ,g n } is also independent? Explain why.