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

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Find the determinant of

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Compute

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

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Suppose What is

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Show that the following vectors are linearly dependent

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

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

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

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

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Find an orthonormal basis for the image of A.

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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.

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Given and Calculate q 3

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

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Determine the eigenvalues and the eigenvectors of A.

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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

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

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Find the area of the parallelogram spanned by a and b

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

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

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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