Linear Algebra Study Notes

LINEAR ALGEBRA NOTES

Course Information

• Course Code: 015403525
• Course Title: EL 18
• Institution: University of Mines and Technology
• Faculty: Computing and Mathematical Sciences
• Department: Mathematical Sciences
• Course Number: EL 171

Problem Set

Problem 1: Solve for z

• Given: $z = -2 + (2 ext{√}3)i$
• Required Form: Express the roots in the form: $r( ext{cos} heta + i ext{sin} heta)$
• Solution Steps:
  • Identify the modulus, $r$:
    • $r = ext{√}((-2)^2 + (2 ext{√}3)^2)$
    • $r = ext{√}(4 + 12)$
    • $r = ext{√}(16) = 4$
  • Identify the argument, $heta$:
    • $heta = ext{tan}^{-1}\frac{2 ext{√}3}{-2}$
    • Since the point is in the second quadrant, adjust:
    • $heta = ext{π} - ext{tan}^{-1}( ext{√}3) = ext{π} - \frac{ ext{π}}{3} = \frac{2 ext{π}}{3}$
  • Final expression:
    • $z = 4 \bigg( ext{cos}\frac{2 ext{π}}{3} + i ext{sin}\frac{2 ext{π}}{3}\bigg)$

Problem 2: De-Moivre's Theorem Application

• Task: Show that $an(40) = ext{exact roots of the equation}$
  $4 ext{√}3 - 6 - 4 ext{√}3 + 1 = 0$
• Solution Steps:
  • Simplifying the expression given:
    • Combining like terms: $4 ext{√}3 - 4 ext{√}3 - 6 + 1 = -5$
    • Equation simplifies to:
    • $-5 = 0$ which leads to an inconsistency.
  • Investigate values of tangent at related angles:
    • Utilizing De-Moivre's theorem:
      • $ext{(cos} heta + i ext{sin} heta)^n = ext{cos}(n heta) + i ext{sin}(n heta)$
    • For $40^ ext{o}$:
      • Consider multiples of $tan(40^ ext{o})$ related to cyclical patterns in trig functions.

Problem 3: Binomial Expansion

• Given: $(1+x)^n = P<em>0 + P</em>1 x + P<em>2 x^2 + P</em>3 x^3 + …$
• Prove that:
  • $an(0)-4 = \frac{1-6 an^2(0)}{4}$
  • i. $P = P<em>2 + P</em>1 = 2^2 ( ext{cos}(17t))$
  • ii. $R = P<em>3 + P</em>t = \frac{2^2 ( ext{sin}(t))}{4}$
• Implications of the Expansion:
  • Relate the binomial coefficients to combinatorial interpretations, such as selection of elements from a set.