Review of Factorials. use the skills covered in the Brief Review on p. 458 to evaluate the following quantities without using the factorial key on your calculator (you may use the multiplication key). Show your work. 11. 6! 12. 12! 13.
15.
17.
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21.
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Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
- 11. Positive Definiteness Prove that a matrix A is positive definite if and only if all its eigenvalues are positive.arrow_forward21. Change of Basis Prove that the matrix representation of a linear transformation T : V → V depends on the choice of basis in V. If P is a change of basis matrix, show that the transformation matrix in the new basis is P-¹AP.arrow_forward14. Projection Matrices Show that if P is a projection matrix, then P² = P. Find the projection matrix onto the subspace spanned by the vector (1,2,2)T.arrow_forward
- 4. Diagonalization Prove that a square matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. • Determine whether the following matrix is diagonalizable: [54 2 B = 01 -1 3arrow_forward8. Determinants • • Prove that the determinant of a triangular matrix is the product of its diagonal entries. Show that det(AB) = det(A)det(B) for any two square matrices A and B.arrow_forward15. Tensor Products • • Define the tensor product of two vector spaces. Compute the tensor product of (1,0) and (0, 1) in R². Discuss the role of tensors in multilinear algebra and provide an example of a second-order tensor.arrow_forward
- 20. Numerical Methods • Describe the QR decomposition method and explain its use in solving linear systems. • Solve the following system numerically using Jacobi iteration: 10x+y+z = 12, 2x+10y+z = 13, 2x+2y+10z = 14.arrow_forward1. Vector Spaces • Prove that the set of all polynomials of degree at most n forms a vector space over R. Determine its dimension. • = Let VR³ and define a subset W = {(x, y, z) Є R³ | x + y + z = 0}. Prove that W is a subspace of V and find its basis.arrow_forward24. Spectral Decomposition Explain the spectral decomposition of a symmetric matrix and its applications. • Compute the spectral decomposition of: A = 5 4arrow_forward
- 3. Eigenvalues and Eigenvectors • Find the eigenvalues and eigenvectors of the matrix: 2 1 A = = Prove that if A is a symmetric matrix, then all its eigenvalues are real.arrow_forward25. Kronecker Product Define the Kronecker product of two matrices. Prove that the Kronecker product of AЄ Rmxn and B ERP is a block matrix in Rmpxnq • Compute the Kronecker product of: A [1 2 3 4 ' B [ ].arrow_forward10. Singular Value Decomposition (SVD) Explain the Singular Value Decomposition (SVD) of a matrix and its applications. • Compute the SVD of the matrix: Darrow_forward
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