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    Linear Algebra

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    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 Frequency Count

    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

    Transcendental Equation, Positive-Definite and Orthonormal

    Solve the eigenvalue problem as follows: Let U = ... be a two-component vector whose first component is a twice differentiable function u(x), and whose second component is a real number u1 Consider the corresponding vector space H with inner product Let S C H be the subspace .... and let .... The above eigenvalue

    Differential Operators: Eigenvalues and Eigenfunctions

    Please see the attached file for the fully formatted problems. Let L = with boundary conditions u(0) = 0, u'(O) = u(1) ,so that the domain of L is S = {u Lu is square integrable; u(0) = 0, u'(O) = u(1)}. (a) For the above differential operator FIND S* for the adjoint with respect to (v,u) =S 1-->0 v-bar u dx and compare S

    Eignevalues and Eigenvectors: Example Problem

    Please see the attached file for the fully formatted problems. The Fourier transform, call it F, is a linear one-to-one operator from the space of square-integrable functions onto itself. (In fact, we also know that F is an "isometric" mapping, but we will not need this feature in this problem). Indeed, Note that here x an

    Linear algebra

    Determine the characteristic values of the given matrix and find the corresponding vectors: [ 2 -2 1 ] [ 1 -1 1 ] [ -3 2 -2 ]

    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

    Geometric Interpretation of Subspaces on Planes

    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.

    Linear Algebra And Differential Equations: Real Vector Space

    Determine if the given set constitutes a real vector space. The operations of "multiplication by a number" and "addition" are understood to be the usual operations associated with the elements of the set: The set of all elements of R^3 with first component 0

    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|

    Diagonal matrix

    For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). The matrix A = [T]_E is similar to a diagonal matrix D = [T]_F. Write the diagonal matrix D, and demonstrate that it is indeed similar to A by producing the appropriate non-singular matrix and its i

    Change of Basis: Eigenvectors

    For the problem, refer to the linear transformation T: R^3 --> R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Write the change of basis matrix K from the basis F of R^3 which consists of the eigenvectors of T to the standard basis E for R^3.