Matrix Representation : Orthogonal Transformation

Please see the attached file for full problem description. 1. Show that the 3 x 3 matrix P= [ 1/2 -1//2 0 ] [ 0 0 1 ] [ 1/2 1/2 0 ...continues

Matrix Representation : Orthogonal Transformation and Determinant

Please see the attached file for full problem description. Write a proof for the following statement: If P is an n x n orthogonal matrix, then det(P) = 1 or –1. Show work. Help: det: is the determinant

Eigenvalues of a Transformation

The linear operator T: R^3 -> R^3 defined by T(x_1, x_2, x_3) = (x_1 - 3x_3, x_1 + 2x_2 + x_3, x_3 - 3x_1). Find the eigenvalues of the transformation T. Show work. (See attachment for the full question.)

Eigenspaces; transformations

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Transformations : Diagonalization of Matrices

Please see the attached file for full problem description. The linear operator T: R^3 R^3 defined by T(x_1, x_2, x_3) = (x_1 - 3x_3, x_1 + 2x_2 + x_3, x_3 – 3x_1). Determine whether or not there is a basis F for R^3 relative to which the transformation T can be represented by a diagonal matrix D=[T]_F. If there is, ...continues

Proof : Diagonalization of Matrices

Please see the attached file for full problem description. Write a proof for the following statement: If A is an n x n upper triangular matrix with no two diagonal elements the same, then A is similar to a diagonal matrix. Show work.

Findin the Equation of a Reflecting Line

Determine if the following orthogonal matrix represents a rotation or a reflection of the plane with respect to the standard basis. Find the equation of the reflecting line. - - |3/5 4/5 | |4/5 -3/5 | - -

Idempotent linear transformation

A linear transformation L:V->V is said to be idempotent if L dot L = L. If L is idempotent, show that there exists a basis S={a1,a2,...,an} for V such that L(ai)=ai for i= 1,2,...,r and L(aj) = 0v for j= r+1,...,n, where r= p(L). Describe the matrix representing L with respect to the basis S.

Nilpotent transformation

Consider the transformation N: V->V. Let g be a vector such that N^k-1 does not equal 0, but N^k = 0. First show that the vectors g,N(g),N^2(g),..,N^k-1(g) are linearly independent, and then (assuming V has dimension n) If N is nilpotent of index n, show that the set S= {g, N(g), N^2(g),...,N^n-1(g)}is a basis for V. Describe th ...continues

Linear algebra; linear dependence

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