Linear Algebra, 2020a

400 Chapter Five. Similarity
1.8 Example Because of uniqueness we know, without multiplying them out, that
(x + 3) 2 (x 2 + 1) 3 does not equal (x + 3) 4 (x 2 + x + 1) 2 .
1.9 Example By uniqueness, if c(x) =m(x)·q(x) then where c(x) =(x−3) 2 (x+2) 3
and m(x) =(x − 3)(x + 2) 2 , we know that q(x) =(x − 3)(x + 2).
While x 2 +1 has no real roots and so doesn’t factor over the real numbers, if we
imagine a root — traditionally denoted i, so that i 2 + 1 = 0 — then x 2 + 1 factors
into a product of linears (x−i)(x+i). When we adjoin this root i to the reals and
close the new system with respect to addition and multiplication then we have
the complex numbers C = {a + bi | a, b ∈ R and i 2 =−1}. (These are often
pictured on a plane with a plotted on the horizontal axis and b on the vertical;
note that the distance of the point from the origin is |a + bi| = √ a 2 + b 2 .)
In C all quadratics factor. That is, in contrast with the reals, C has no
irreducible quadratics.
ax 2 + bx + c = a · (x
− −b + √ b 2 − 4ac
2a
) ( −b − √ b 2 − 4ac)
· x −
2a
1.10 Example The second degree polynomial x 2 + x + 1 factors over the complex
numbers into the product of two first degree polynomials.
(
x −
−1 + √ −3
2
)( −1 − √ √
−3) ( 1 3
x − = x −(−
2
2 + 2 i))( x −(− 1 √
3
2 − 2 i))
1.11 Theorem (Fundamental Theorem of Algebra) Polynomials with complex coefficients
factor into linear polynomials with complex coefficients. The factorization
is unique.
I.2 Complex Representations
Recall the definitions of the complex number addition
and multiplication.
(a + bi) +(c + di) =(a + c)+(b + d)i
(a + bi)(c + di) =ac + adi + bci + bd(−1)
=(ac − bd)+(ad + bc)i
2.1 Example For instance, (1 − 2i) +(5 + 4i) =6 + 2i and (2 − 3i)(4 − 0.5i) =
6.5 − 13i.