University:
Massachusetts Institute of Technology
Course:
18.06 | Linear Algebra
Academic year:

2024
Views:

256

Pages:

2
Author:

Mckenna Welch

Question 1 Whats 5 times 5 and the quadratic formula to 24 3 Answer The first answer: 25 The second: 3x²-24=0 Question 2 A lattice point in the xy-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the hyperbola x² - y² = 17? I think the answer should be 4, because x² - y² = (x+y)(x - y) = 17. 17 has 4 factors: 1,17, -1, and -17. But I don't know if these numbers actually work. Answer Step 1 x + y = 17 and x - y = 1 => 2x = 18 => x = 9 => y = 8 x + y = 1 and x - y = 17 => 2x = 18 => x = 9 => y = -8 Step 2 x + y = -17 and x - y = -1 => 2x = -18 => x = -9 => y = -8 x + y = -1 and x - y = -17 => 2x = -18 => x = -9 => y = 8. Question 3 Why is sum of roots of quadratic equation power an integer coefficients of the expansion -f' -x -? f(x) 1 Calculate the ∑ aⁿ n∈Z where a is the roots of a quadratic equation in the form f(x) = x² + ax + b. Answer Let f(x) be a monic polynomial of degree d with all roots a₁, ..., a_d simple. In other words, d f(x) = (x − a₁) (x − a₂) ... (x - a_d) = Π (x - α_k). Then Inf(z) = ∑ ln (z - α_k) k=1 and with g(z) = -f'(z)/f(z) we observe -z f'(z)/f(z) = -z d/dz ln f(z) = -z * ∑(d/dz ln(z - α_k) = ∑(α_k)/(z-α_k) = 1/1-α_k/z Now use the geometric series 1 / (z - α_k) = 1/z * 1/(1 - α_k/z) = 1/z * ∑ (α_k/z)ⁿ = ∑(α_k)ⁿ/z^(n+1) so that by comparing coefficients of equal powers of z in the power series, we conclude an = ∑ (α_k)^(n+1). Question 4 Can this be an onto function? A function f: R → R, where R is set of real numbers, is defined by f(x) = (ax² + 6x - 8) / (a + 6x - 8x²) Find the interval of values of a for which f is onto. Justify your answer. Answer The question is poorly worded. They do not want the domain to be the set of real numbers, they want it to be {x ∈ R : a + 6x – 8x² ≠ 0}. In calculus and analysis it's common for functions not to be defined everywhere and many times the phrase "real valued function" means the domain is an appropriate subset of the real numbers. The proper term for this is a "partial function" but since these are so common in this area of mathematics, it's common to just call these "functions." Whether it's still appropriate to write "f : R → R" I would argue no. I've always seen f : D ⊂ R → R where D is an unspecified set but is assumed to be the largest subset of R for which the definition of f makes sense. It would seem that whoever wrote this textbook have a different opinion than me. Question 5 Factor a quadratic when the leading coefficient is not equal to 1 and you can't factor by grouping? If one has a quadratic, for example 5x² - 10x - 2, which has real roots which via the quadratic equation are (5 ± √35) / 5, can you find its factored form. As I understand it 5x² - 10x - 2 ≠ (x-root1)(x-root2)? How can you turn this type of quadratic into a factored form? Answer Take the coefficient of the squared term (In this quadratic the coefficient is 5). Multiply this coefficient multiply it by (x - root₁) and (x - root₂) where root₁ and root₂ are the two factors of the quadratic polynomial (In this quadratic the two roots are (5 + √35)/5 and (5 - √35)/5). The final solution being: 5(x - (5 + √35) / 5)(x - (5 - √35) / 5)