Linear Algebra: Matrix of Transformation

Are the following examples linear transformations from p3 to p4? If yes, compute the matrix of transformation in the standard basis of P3 {1,x,x^2} and P4 {1,x,x^2,x^3}. (a) L(p(x))=x^3*p''(x)+x^2p'(x)-x*p(x) (b) L(p(x))=x^2*p''(x)+p(x)p''(x) (c) L(p(x))=x^3*p(1)+x*p(0)

Linear Algebra: Vector Spaces

Consider R2 with the following rules of multiplications and additions: For each x=(x1,x2), y=(y1,y2): x+y=(x2+y2,x1+y1) and for any scalar alpha, alpha*x=(alpha*x1, alpha*x2) Is it a vector space, if not demonstrate which axioms fail to hold. Also, show that Pn- the space of polynomials of order less than n is a vector spac ...continues

Linear Algebra; Finding a matrix term

Consider a subspace S of C[-pi,pi] formed by Span of the following functions sin(nx), n=1,2...(infinite dimensional subspace). Consider a linear operator (Schrodinger) acting on a function f(x) from S L(f)=d2f/dx2+x^2*f(x) This operator can be written as an infinite-dimensional matrix in the basis sin(nx). Find the (k,l) ...continues

Linear Algebra: Linear Mapping

Consider the following linear mapping from C[-pi,pi] into itself: L(f)=integral from -pi to pi of G(x),h(y),f(y)dy for any function f(x) in C[-pi,pi]. Here G(x), H(x) are given continuous functions. Find a function f such that L*f=lambda*f for some lambda and find the value of lambda. This is a generalization of the notion ...continues

Veryfying Solutions to Linear Equations

THE ORDERED PAIR (-2,-23/5) IS A SOLUTION TO WHICH LINEAR EQUATION? 4X+5Y= -2 Y=4/5X-3 4X= -5Y+2 OR NONE OF THESE?

In this problem, you are asked to find a new basis in which the matrix A is diagonalized.

Suppose that the matrix A is a linear operator on R(2) represented in the standard basis {i,j}, by the matrix mat(A)= [-1 3;2 4]. Find a new basis in which A is diagonalized.

This problem is a proof involving eigenvalues and eigenvectors.

If x is an eigenvector of the nxn matrix A, prove that bx is also an eigenvector.

Show that a matrix is symmetric and idempotent

Let X be a txk matrix whose columns are linearly independent. Let M=I-X(X'X)^(-1)X'. Show that M is symmetric and idempotent.

Ratio and Proportion: Cat-Mice Problem

Five cats can catch 5 mice in 5 minutes. At this rate, how many cats are needed to catch 100 mice in 100 minutes? (Hint: it is not 100.)

Equation: GCD and Mod

Find the least number N such that 1024N-11529=15625x where N and x are non-negative integers