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Linear Algebra

Linear Algebra : Wronskian

Compute the Wronskian of the given set of functions, then determine whether the function is linearly dependent or linearly independent: x^2 - x, x^2 + x, x^2, all x

Linear algebra

Show that the set of all elements of R^3 of the form (a + b, -a, 2b), where a and b are any real numbers, is a subspace of R^3. Show that the geometric interpretation of this subspace is a plane and find its equation.

Subgoups : Indicies

Note: C means set containment (not proper set containment), |G : K| means index of subgroup K in G, and G # K means K is a normal subgroup of G question: Let K C H C G be groups, where K # G and |G : K| is finite. Show that |G/K : H/K| is also finite and that |G/K : H/K|=|G : H|

Finding eigenvalues

Let Tn = (tij) denote the nxn matrix such that for each index i, tji = a and tij = b for j not = to i. Verify that Tn = (a-b)In + bEn where En is the nxn matrix of all 1's. Find the determinate and the eigenvalues of Tn.

Solving for Eigenvalues

A certain 4x4 real matrix is known to have these properties: 1) Two of the eigenvalues of A are L1= 3 and L2= 2. 2) The number 3 is an eigenvalue of the matrix A + 2I. 3) det. A = 12. What are the other 2 eigenvalues of A and what is the characteristic polynomial of A, At, and A-1?

Eigenvector Estimation : Inverse Power Method

Please see the attached file for the fully formatted problems. Use the inverse power method to estimate the eigenvector corresponding to the eigenvalue with smallest absolute value for the matrix -1 -2 -1 A= -2 -4 -3 2 2 1 where X0= [1,1,-1]. In finding A-1 use exact arithmetic with fractions. ln applyi

Normal subgroups

Let X be a nonempty subset of a group G. If G = <X> and H is a subgroup of G, show that H is the normal subgroup of G if and only if x^-1Hx contained in H for all x belonging to X. ALSO show that <X> is normal in G if and only if gXg^-1 contained in <X> for all g belonging to G.

Subgroups

"C" means set containment (not proper set containment) and "T" means intersection of sets If H and K are subgroups of a group and |H| is prime, show that H C K or H T K = {1}

Cosets/subgroups

Let G = RxR (R is the real numbers) with addition (x,y) + (x', y') = (x+x', y+y'). Let H be the line y=mx through the origin: H = {(x,mx)such that x belongs to R (R is real numbers). Show that H is a subgroup of G and describe the cosets H + (a,b) geometrically.

Cosets

If H is a subgroup of G, define a mapping $ from the right cosets of H to the left cosets by $(Ha) = a^-1H. Show that $ is a (well defined) bijection.

Homomorphisms

If $:G->G1 is a homomorphism, show that K = the set of g belonging to G given that $(g)=1 is a subgroup of G (called the kernel of $)

Signals - System Properties

I have difficulty in determining whether the signals are memoryless or causal. Please see the attached file for full problem description.

Digital signal systems problem

The book I am using is "Digital Signal Processing" (Third Edition) by Prokis and Manolakis. It is question 2-45 on page 144. Consider the system described by the difference equation: y(n)=a*y(n-1)+b*x(n) determine b in terms of a so that THE SUMMATION OF h(n)=1 The limits of the summation are

Subgroups

If X is a nonempty subset of a group G, let <X>={x1^(k1),x2^(k2)...xm^(km)|m>=1, xiEX and kiEZ for each i}. a) show that <X> is a subgroup of G that contains x. b) show that <X>C=H for every subgroup H such that XC=H. Thus <X> is the smallest subgroup of G that contains X, and is called the subgroup generated by X. note:

Linear algebra

Q.63. For each i,j &#8805; 0, define P (i,j) as follows : P (0,0) = 0, and , for (i,j) &#8800; (0,0) , P(i,j) is the least integer &#8805; 0 that is not equal to P (k,l) for any (k,l) with k < i and l = j , or with k = I and l < j. Find P (2987,6592).

Linear velocity

Find the linear velocity of a point on the edge of a drum rotating 52 times per minute. The diameter of the wheel is 16.0in. Please show me all the steps thank you

Linear Algebra

For what value of the parameter b will the following system of equations fail to have a unique solution? (HINT - Do not attempt to actually solve the equations!!!!) x+2by-z = 2 2bx+3y-bz = 3 x+2y+z = 0