Math Review

1. Math Review CS474/674 – Prof. Bebis

2. Math Review • Complex numbers • Sine and Cosine Functions • Sinc function • Vector Basis • Function Basis

3. Complex Numbers • A complex number x is of the form: α: real part, b: imaginary part • Addition: • Multiplication:

4. Complex Numbers (cont’d) • Magnitude-Phase (i.e., vector) representation: Magnitude: Phase: φ Magnitude-Phase notation:

5. Complex Numbers (cont’d) • Multiplication using magnitude-phase representation • Complex conjugate • Properties

6. Complex Numbers (cont’d) • Euler’s formula • Properties 2j

7. Sine and Cosine Functions • Periodic functions • General form of sine and cosine functions: y(t)=Asin(αt+b) y(t)=Acos(αt+b)

8. Sine and Cosine Functions (cont’d) Special case: A=1, b=0, α=1 period=2π π 3π/2 π/2 π 3π/2 π/2

9. Sine and Cosine Functions (cont’d) • Changing the phase shift b: Note: cosine is a shifted sine function:

10. Sine and Cosine Functions (cont’d) • Changing the amplitude A:

11. Sine and Cosine Functions (cont’d) • Changing the period T=2π/|α|: Asssume A=1, b=0: y=cos(αt) α =4 period 2π/4=π/2 shorter period higher frequency (i.e., oscillates faster) frequency is defined as f=1/T Alternative notation: cos(αt) or cos(2πt/T) or cos(t/T) or cos(2πft) or cos(ft)

12. Sinc function • The sinc function is defined as: • In image processing, we use the normalized sinc function: The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the un-normalized sinc function has a value of π). Also, it crosses the x-axis at integer locations.

13. Vectors • An n-dimensional vector v is denoted as follows: • The transpose vTis denoted as follows:

14. Dot product • Given vT= (x1, x2, . . . , xn) and wT= (y1, y2, . . . , yn), their dot product defined as follows: (scalar) or

15. Linear combinations of vectors • A vector v is a linear combination of the vectors v1, ..., vk if: where c1, ..., ck are scalars • Example: any vector in R3 can be expressed as a linear combinations of the unit vectors i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1)

16. Linear dependence • A set of vectors v1, ..., vkare linearly dependent if at least one of them is a linear combination of the others. (i.e., vj does not appear at the right side)

17. Linear independence • A set of vectors v1, ..., vkis linearly independent if implies Example:

18. w Space spanning • A set of vectors S=(v1, v2, . . . , vk ) span some space W if every vector in W can be written as a linear combination of the vectors in S • Example: the vectors i, j, and k span R3

19. Vector basis • A set of vectors (v1, ..., vk) is said to be a basis for a vector space W if (1) (v1, ..., vk)are linearly independent (2) (v1, ..., vk)span W • Standard bases: R2 R3 Rn

20. k Orthogonal/Orthonormal Basis • A basis with orthogonal/orthonormal basis vectors. • Any set of basis vectors (x1, x2, . . . , xn) can be transformed to an orthogonal basis (o1, o2, . . . , on) using the Gram-Schmidtorthogonalization.

21. Uniqueness of Vector Expansion • Suppose v1, v2, . . . , vn represents a base in W, then any v єW has a unique vector expansion in this base: • The coefficients of the expansion can be computed as follows:

22. Basis Functions • We can compose arbitrary functions x(t) in “function space S” as a linear combination of simpler functions: • The set of functions φi(t) are called the expansion set of S. • If the expansion is unique, the set φi(t) is a basis.

23. Basis Functions (cont’d) • Example: polynomial basis functions φi(t) = ti

24. Orthogonal/Orthonormal Basis • φi(t)are orthogonal in some interval [t1,t2] if: • For complex valued basis sets:

25. Coefficients of Expansion • The coefficients of the expansion can be computed as: