Determinants Bases, linear Indep., etc Gram-Schmidt Eigenvalue and Eigenvectors Misc. 200 400 600 800.

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Presentation transcript:

Determinants Bases, linear Indep., etc Gram-Schmidt Eigenvalue and Eigenvectors Misc

Find the determinant of

Compute

The axiomatic definition of the determinant function includes three axioms. What are they?

Suppose What is

Show that the following vectors are linearly dependent

Find the rank of A and the dimension of the kernel of A

Find a basis for the kernel of A and for the image of A

Find the equation of a plane containing P, Q, and R

Let Q be an orthonormal basis for the matrix S. Find the matrix of the orthogonal projection onto S.

Find an orthonormal basis for the image of A.

For some matrix A, there exists Q and R as given s.t. A=QR. Solve the least squares problem Ax=b for the given b.

Given and Calculate q 3

Given A and the correspond char. polynomial, find the eigenvalues and eigenvectors of A.

Determine the eigenvalues and the eigenvectors of A.

Given A and the char. polynomial, determine: 1.The eigenvalues of A 2.The Geometric and algebraic multiplicities of each eigenvalue 3.Is it possible to find D and V such that A = VDV -1 ? Justify your answer

Find a diagonal matrix D and an Invertible matrix V such that A=VDV -1 Also calculate A 8.

Find the area of the parallelogram spanned by a and b

What are two methods you know for calculating the solution to a least squares regression problem which use the Gram-Schmidt QR factorization?

Find the area of the triangle determined by the points (0,1), (2,5), (-3,3)

In the theory of Markov Chains, a stationary distribution is a vector that remains unchanged after being transformed by a stochastic matrix P. Also, the elements of the vector sum to 1. Determine the stationary distribution of