Q1: Let V1 =(1,1,1), V2 =(-1,1,0) and V3 =(-1,0,1).
Show that{V1,V2,V3} is basis of .
Apply the Gram-Schmidt orthogonalization process to construct an orthogonal basis of using {V1,V2,V3}.
Q2: Solve the following.
Write the matrix of T1, the transformation in obtained by doing 60° counterclockwise rotation about the x-axis.
Write the matrix of T2, the transformation in obtained by doing 90° counterclockwise rotation about the z-axis.
Compute T2(T1(1,0,1)).
Q3: The smallest subspace of containing the vectors (2,-3,-3) and (0,3,2) is the plane whose equation is ax + by + 6z=0. Find the values of a and b. show all the steps.
Q4: Compute the Cholesky decomposition (L L^T) of the matrix (when needed, round your answers to two decimals):
C=
Q5: Let B=
Show that B^tB = .
Find the singular values of the matrix B.
Find a singular value decomposition (SVD) of the matrix B.
Q6: let A= .
Show that the characteristic polynomial of A is –(λ-2)(λ-1)².
Find the eigenvalues of A and give their algebraic multiplicities.
Find the dimension of A and compare the algebraic and geometrical multiplicities of each eigenvalue.
Show that A is diagonalizable, that is fond a matrix p and a dingonal matrix D such that A= PDP^-1.
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