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Exercises:. Let Mn(F) denote the set of all square matrices of order n, over a field of scalars, F. Prove that Mn(F) forms a vector space over F with respect to matrix addition and scalar multiplication. Prove that the determinant of an upper triangular matrix is the product of its diagonal element Let A be a matrix where A M n (F ) , such that det A ≠ 0. Prove that if ( A) then 1 ( A 1 ) 0 1 Given the square matrix A M 2 ( Z 5 ) , where A , compute: 1 0 (i ) det A (ii ) the eigenspace for each ( A) . (iii ) the Jordan Canonical Form If A, B M n ( F ) where A ~ B (specially, A = S-1BS, where det S 0 ), then the following properties hold: (i) det A = det B; (ii) if A is diagonalizable, then so is B; (iii) if x is an eigenvector of A, then Sx is an eigenvector of B. If A, B M n ( F ) are simultaneously diagonalizable, then they have a common eigenvector. Prove that the eigenvalues of a real n-dimensional symmetric matrix are real, and all eigenvectors corresponding to each eigenvalue are orthogonal. Solve the non-homogeneous ODE 3 1 1 dX AX F (t ) , in which A 2 0 1, F (t ) (0,0, e 2t ) T dt X(0) (1,1,1) T 1 1 2 Apply linear transformation, transform y=4x12+6x1x2-4x22-20x1+10x2-8 into y=a0+a1t12+a2t22 ; And find the value of a0,a1,a2. 2 1 1 Given A 1 4 3 , try to find eigenvalue s and eigenvecto rs, then diagonalize A 1 3 4