find the orthogonal projection of v onto the subspace W spanned by the vectors ui . (You

Finite Math 1­17­18 Lecture  A set is a well­definted collection of objects  Example: {New England states} = {MA, VT, NH, CT, RI, ME}  {Midwest states} = not a well­defined set, people have different ideas about what is Midwest  ways to represent sets o via an unambiguous rule: P = prime numbers between 2 and 20 o via a list, eg P = {2, 3, 5, 7, 11, 13, 17, 19} o “…” notation: {2, 4, 6, 8, 10…2020} is all even numbers between 2 and 2020  can also be used to represent infinite sets, eg {1, 4, 9, 16, 25, 36…} means infinite set of all square numbers o set builder notation: {x|x is an even number between 2 and 2020 inclusive}={the set of x such that x is…}  more notation o “7  P” reads “7 is an element of P” o “8 (arrow with a line through it) P” reads “8 is not an element of P”  relations between sets o subsets: A is a subset of B if all elements of A are also elements of B  A=states with one syllable name=Maine  B=New England States  So BA because Maine is a New England state  C=states with 2 word names, so the same relationship cannot be made because there are two word states not in New England  Q: How many subsets does {1, 2, 3} have o A: subsets of {1, 2, 3}= {2}, {1}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}  Empty set: a set with no elements, denoted by  o {x|x is a prime number bigger than 11 and less than 13} is an empty set  back to