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Linear Algebra

Linear Algebra Question

1. Let V be a vector space of odd dimension (greater than 1) over the real field R. Show that any linear operator on V has a proper invariant subspace other than {0}.

Perturbed Linear System

Consider the perturbed linear system x' = (A + eB(t))x, x is an element of R^n, where A is a constant matrix, B is a bounded continuous matrix valued function, and e is a small parameter. Assume that all eigenvalues of A have non-zero real part. 1) Show that the only bounded solution of the system is 0. 2) If A ha

Systems of Equations : Applications to Perimeter of a Rectangle

Write as a system of 2 equations in 2 unknowns. Solve each system by substitution and show each step. Perimeter of a rectangle : the length of a rectangular swimming pool is 15 feet longer than the width. If the perimeter is 82 feet, what are the length and width?

Systems of Equations : Five Word Problems

There are hens + rabbits. The heads = 50 the feet = 134. How many hens & how many rabbits ? Flies + spiders sum 42 heads and 276 feet. How many of each class? J received $1000 and bought 9 packs of whole milk & skim milk that totalled $960 - How many packs bought of each kind? A number is composed of two integers and its sum

Cryptogram

Solve the following cryptogram doing the following steps: 1. frequency count 2. do you think it is monoalphabetic substitution, polyalphabetic subsitution, or transposition? 3. Is this a clear decision? 4. Solve based on the above info. DVOLL PULID ZIWGL ZDLIO WULFM WVWFM WVWFK LMULF IVHHV MGRZO SFNZM UIVVW LNHGS VUR

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.