A Function of Years

D(x) is the number of days in the year x. Leap years have 366 days. Others have 365 days. D(1980)= D(1950)= D(1776)= This is three of our problems. How do we calculate these equations? The above info is all that is given.

Linear alegbra

Consider the following elements of the vector space P3 of all polynomials of degree less than or equal to 3. p(x)= x-1, q(x)=x+x2, r(x)= 1+x2-x3 Do these three polynomials form a basis for P3?

Linear Alegbra and Vectors : Linear Dependence

Determine whether or not the following set of vectors in R4 is linearly independent? u=(1,0,3,0) v=(2,1,1,3) w=(1,2,-7,6)

Linear equation

Consider the equation Ax=b, with a=(1 1 a) (1 -1 1) (2 -1 -1) b=(6+b) ( b ) ( b ) for which values of a,b this system has no solutions? infinitely many solutions? unique solution? if possible, find the solution x explicitly in terms of a,b.

Matrix subspace

Consider the matrix a=(1 1 2 1 2 -1 3 2 1 5 5 2) Find N(A), R(A), N(A^T),R(A^T). Show that the fundamental subspace theorem holds: N(A^T)=R(A)^(upside down T), N(A)=R(A^T)^(upsidedown T). Hint: Notice that the fourth row is the sum of the first three rows.

Linear algebra

Suppose S is a linear space defined below. Are the following mappings L linear transformations from S into itself? If answer is yes, find the matrix representations of formations (in standard basis): (a) S=P4, L(p(x))=p(0)+x*p(1)+x^2*p(2)+X^3*p(4) (b) S=P4, L(p(x))=x^3+x*p'(x)+p(0) (c) S is a subspace of C[0,1] formed by ...continues

Linear Algebra : Orthogonal Projection

Consider vector space C[0,1] with scalar product: for any two functions f(x), g(x) (f,g)=integral from 0 to 1 of f(x)g(x)xdx. Find the orthogonal projection p of e^x onto x. Also find the norms of e^x and x.

Linear Algebra : Norms

For any x=(x1,....,xn), let us try to define a norm two ways. Consider (a) ||X||1=summation |Xi| from i=1 to n (b) ||X||b=summation |xi-xj| from i,j=1 to n Does either one of these formulas define a norm? If yes, show that all three axioms of norm hold. If no, demonstrate which axiom fails.

Linear Algebra: Eigenvalues

Find eigenvalues and eigenvectors of the matrix A=(2 1 9 2) By transforming the matrix in the basis of eigenvectors, show explictly that the matrix can diagonalized in the eigenvector basis.

Determinant of the Van der Monde Matrix

The Vandermonde matrix is defined the following way. Suppose x1,x2,...xn are n numbers. Form the nxn matrix: A=(1 x1 x1^2 ... x1^(n-1) ) (1 x2 x2^2 ... x2^(n-1) ) (... ) (1 xn xn^2 ... xn^(n-1) ) Find determinant A.