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.

Vectors and Linear Algebra

Compute the following vector quantities: a. (i-j-k)x(2i+3j-k) b. (3i-j-k).[(i-j)x(i+j-k)]

Relations between Non-Parallel Planes

Let P1 and P2 be two dimensional planes that are not parallel. If these planes are contained in R3 then they must intersect in a line. Prove that if they are contained in R4 instead then they can intersect either along a line, or at a single point. (HINT)- A two dimensional plane in R4 is determined by two equations in four unkn ...continues

Linear Alegbra : Systems of Equations

Find the solution space of the following system of equations. x1-x2+4x3-x4=2 2x1 -2x3+4x4=4 2x1-x2+3x3+x4=4 x2-5x3+2x4=0

Linear Alegbra : Vector Space

Let V= (x,y) in R2{y=3x+1} with addition and multiplication by a scalar defined on V by: (x,y)+ (x',y')= (x+x',y+y'-1) k(x,y)=(kx,k(y-1)+1) Given that with these definitions, V satisfies vector space axioms 1,2,3,6,8,9,and 10 determine whether or not V is a vector space by checking to see if axioms 4,5,7,are also satisfied.

Vectors

Suppose that a "skew" product of vectors in R2 is defined by (u,v)=u1v1-u2v2 Prove that (u,v)squared >equal too (u,u)(v,v). (NOTE; This is just the reverse of the Cauchy- Schwartz inequality for the ordinary dot product.)

Linear alegbra and vectors

Use the geometric method of linear programming to maximize the objective function f(x,y)=3x-6y subject to the constraints. x>= 1 x-y<= 3 2x+y>= 6 2x+y<= 8