A vector space is a set of objects, called vectors, along with two operations, called addition and scalar multiplication, such that the sum of any two vectors and the scalar multiple of any vector is also a vector. The complex numbers form a vector space. Those would be the numbers of the form a+bi where a and b are real numbers. The sum of two complex numbers is defined by (a+bi)+(c+di)= (a+c)+(b+d)i and the scalar multiple of a complex number is defined by k(a+bi)=(ka)+(kb)i. The elements of any vector space satisfy the commutative and associative properties and every vector space has an identity element. Each element then has an additive inverse meaning their sum is the identity. A set of vectors v1,v2,v3,.....,vn is defined to be linearly independent if c1v1+c2v2+c3v3+.....+cnvn=0 implies c1=c2=c3=.....cn=0. An example of a set of linearly independent complex numbers is v1=1+i and v2=2-i. if c1v1+c2v2=0, then (c1+c1i)+(2c2-c2i)=0 which results in (c1+2c2)+(c1-c2)i=0. This... read more
1) a is a power of 3 so it must be odd. b is odd as well. the sum of two odds is always even so the answer is a+b 2) to intersect the square in 8 points the circle would have to have a radius of less than square root of 2 but more than 1. a radius of square root of 2 or more would intersect the square in at most 4 points and the same could be said of a radius of less than or equal to 1. the equation x^2+y^2=5/4 is a circle of radius sqrt(5)/2 and thus satisfies the two conditions 3) the sum of two sides of a triangle must exceed the third so the two equal lengths must be more than 4.5. this means the perimeter must be more than 4.5+4.5+9=18. the answer is III only 4) the first 7 terms are 2,10,24,44,70,102,140. the median is the middle number in an increasing sequence. the answer is 44 5) the common ratio in the sequence is 4 and therefore the nth term is given by 3*4^(n-1). the 30th term is then 3*4^29=864,691,128,455,135,232 6) the first two ordered pairs are... read more
Interested in preparing for the SAT? Here are 10 questions I constructed. You can send me your answers and I will tell you how you did. 1)If a is a power of 3 and b is an odd integer, which of the following must be an even integer? a. a/b b. ab c. a+b d. 2a+b e. a+2b 2)Let (1,1),(1,-1),(-1,1),and (-1.-1) be vertices of a square. Which of the following is an equation of a circle intersecting the square in 8 points? a. x^2+y^2=2 b. x^2+y^2=5/4 c. x^2-y^2=-2 d. x^2-y^2=-5/4 e. x^2+y^2=1 3) An isosceles triangle has a side of length 9 and the other 2 sides are the same length. Which of the following could be the perimeter of the triangle? I.17 II.18 III.19 a. II only b I and II only c. I and III only d. II and III only e. III only 4) Find the median of the first seven terms of the sequence 3n^2-n 5) Given the geometric sequence 3,12,48,192,768,....., what is the 30th term? 6) Given x values... read more
Consider the points (1,3) and (4,5) in the plane. If we start at the point (1,3) and move 3 units to the right and 2 units up we form a right triangle. This means the distance between my 2 points may be computed using the Pythagorean Theorem. The distance would equal the square root of 3 squared plus 2 squared which is square root of 13. Of course we can do this for any 2 points in the plane and indeed extend this definition to 3 dimensions as well. the distance between points (x1,y1,z1) and (x2,y2,z2) is square root of ((x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2). It is from this same definition that we get the equation of a sphere in 3d. A sphere is the set of all points in 3d which are a fixed distance(called the radius) from another point(called the center). If (a,b,c) is the center of a sphere and r is its radius, then the equation of the sphere is given by (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2. Now I present a problem to you. How many spheres go through the point (5,4,1) and... read more
A polynomial is an expression of the form AnX^n + An-1X^(n-1) + An-2X^(n-2) + ..... + A1X + Ao. Examples are 4x^2 + -3x + 7 and -5x^3 + x^2 - 2x. Let's suppose that a polynomial when divided by x-19 has remainder 99 and the same polynomial when divided by x-99 has remainder 19. What is the remainder when that polynomial is divided by (x-19)(x-99)? Let P(x) be the polynomial. Then we are given that P(x)/(x-19) = Q(x) + 99/(x-19) and P(x)/(x-99) = R(x) + 19/(x-99) where Q(x) and R(x) are polynomials. If we subtract the left and right sides of equation 2 from the left and right sides of equation 1 we see that P(x)/(x-19) - P(x)/(x-99) = Q(x) - R(x) + 99/(x-19) - 19/(x-99). The left side simplifies to [P(x)(x-99) - P(x)(x-19)]/(x-19)(x-99) = (xP(x) - 99P(x) - xP(x) + 19P(x))/(x-19)(x-99) = -80P(x)/(x-19)(x-99). The right side simplifies to Q(x) - R(x) + [99(x-99) - 19(x-19)]/(x-19)(x-99) = Q(x) - R(x) + (99x - 99^2 - 19x + 19^2)/(x-19)(x-99) = Q(x) - R(x) + [80x - (99^2 - 19^2)]/(x-19)(x-99)... read more
How do you solve 8/x = 2? Many people will say the answer is 16. But certainly you will agree 8/16 is not 2. The problem is the variable is in the denominator. You fix that by multiplying both sides by x. That is, x*(8/x) = x*2. This results in 8 = 2x, a much easier problem for most people. Then after dividing both sides by 2 we see the correct answer is 8/2 = x = 4. Could we have gotten there more directly? Yes. Look at the original problem 8/x = 2 and the last equation 8/2 = x. The x and the 2 just switched places. The two steps we used will be the same regardless of what numbers we are given. For instance, 14/x = 21 can be solved immediately by switching the x and the 21. The answer is then 2/3. It's called the Van Delden Switch. Yes I named it after myself. Know it, love it, tell a friend, bring it up at parties.