CPS 258 Announcements

1. CPS 258 Announcements • http://www.cs.duke.edu/~nikos/cps258 • Lecture calendar with slides • Pointers to related material • Homework 1 out later tonight, due in 2 weeks

2. Introduction to Vector Calculus and Linear Algebra

3. Positive integers Zero Naturals Negative integers Integers Fractions Rationals Square root (Irrationals) Reals Imaginary Root i Complex Numbers

4. Functions • Constant • Polynomial • linear • quadratic • Rational • Algebraic • Non-algebraic • transcendental

5. Multivariate Functions • F([α,β]) = γ • F(α) =[β,γ] • F([α,β]) = [γ,δ]

6. Euclidean n-space • Euclidean n-space is the space of all n-tuples of real numbers [x1, x2, …, xn]T denoted as Rn

7. Vector space • A Vector space over Rn is a set of vectors for which any vectors x, y, and z and any scalarsα, β, have the following properties: • Commutativity:x+y = y+x • Associativity: (x+y)+z = x+(y+z) • Additive identity: For all x, x+0 = 0+x = x • Existence of additive inverse: For any x, there exists a -x such that x + (-x) = 0 • Associativity of scalar multiplication: α(βx) = (αβ)x • Distributivity of scalar sums: (α+β)x = αx + βx • Distributivity of vector sums: α(x+y) = αx + αy • Scalar multiplication identity: 1x = x

8. Linear Algebra Vector space Rn Linear Transformation T: Rn→ Rm T(u+v) = T(u) + T(v) T(α v) = α T(v)

9. Theorem If T: Rn→ Rm is a linear transformation, there exists an mxn matrix A such that T(x) = A x Conversely, the map x → A x is a linear transformation for any matrix A

10. Matrix Multiplication is linear transformation composition

11. Rank of a matrix Dimension of subspace spanned by The rows The columns The dimension of the range(A) = {y | A x = y}

12. Square Matrices Identity matrix I, the matrix of identity transformation I x = x, or T(x) = x Suppose there exists B such that A B = I Then B is the inverse of A and denoted as A-1 A is called non-singular or invertible A A-1 = I = A-1 A

13. Non-Singularity Theorem If A is nxn matrix, then the following are equivalent • A is invertible • A has rank n • The linear system A x = b has a unique solution for any vector b in Rn

14. Rectangular matrices Theorem: If A is mxn, then rank(A) ≤ min(m,n) When rank(A) = min(m,n), then A is full-rank

15. Transposition The transpose of A, denoted as AT consists of the rows of A as columns (A B)T = BT AT

16. Null Space null(A) = {x | A x = 0}

17. Inner product

18. Orthogonality Two vectors x and y are orthogonal iff xTy = 0

19. The Fundamental Theorem of Linear Algebra For any mxn matrix A, the range(AT) and null(A) are orthogonal vector spaces and their dimensions add to n This is a duality theorem, as it relates a property of A to a property of AT