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Problem 1. Ture or False? (To practice, if false, please modify each wrong statement to correct one. or give some counterexample) (1) For an non invertible n 72 matrix A, The mank of A plus the algebraic multiplicity of eigenvalue O ia equal to the size of A (2) A diagonalizable if and only if the geometric multiplicity and algebraic multiplicity of each eigenvalue are equal. (3) if / a 2 by matrir and frace of A is 0. then the determinant of 4 is not able to be greater than (4) For any matriz M. we can find some invertible P such that PM is the reduced TOW echelon form. (5) Suppose V and W are two linear spaces, and there xa an surjective map from Vto W. then the dimension of W is less than the dimension of V. Problem 2. Find the invertible 3 matrix P. and 5x5 invertible matrix Q such that PMQ is equal to PMQ where M Problem 3. Suppose Vist & dimensional linear space over R with chosen basis e1 e e e4 ) Let W {ae bez cea a-2b-d 2a +6-c-d = = U = {a, ben des a d d a-b-c ce3 (1) Find basa for W and U. (write in terms of €1,e2,ey,es) (2) Find baza for W U (write in terms of (3) Find a basis for wnu (wire in terms of €1 €2 €3 es) (4) For each of the following w. Determine whether it is in subspace W or not. if is in W. find the coordinate of these vectors with respect to your basis. Problem 4. Suppose Vis 3-dimensional linear space with basis (e e2 es ) and T:V V is linear transformation such that Ter And W = {ae + bez+cegla+b-c=0 (1) Find a basis for W (2) Show that W is an invariant subspace of T. (3) With respect to your basis on subspace, find the matrix representation of T/w. where Th means restriction of Ton W. (4) Find the kernel of T/W (5) Find the image of T/w (6) Show that W ker T2 Problem 5. Find an invertible matrix P. such that p-1MP xa an upper triangular matrix. Where M Problem 6. Suppose V M2x2(R) is all2x2 matrices over IR. Consider the linear transformation T: V - V A h Suppose we have the bases: E1 (1) Find the matrix representation of T with respect to the basis (E1 €2 €3 €4 (3) Find the eigenvalues, eigenvectors, the geometric and algebraic multiplicity of each eigenvalue (4) Does this linear transformation diagonalizable? if it is find an eigenbasis. If not. explain why. Problem 7. Let (r41)(542)(243) € C} be a 5-dimensional linear space over C. Substitute by 17 defines linear transformation T V - V (1) Choose a basis for this linear space. (2) Find the matrix representation form of this linear transformation. (3) Find the eigenvalue, eigenvector, the geometric and algebraic multiplicity of each eigenvalue (4) Does this linear transformation diagonalizable? if is find an eigenbasis If not. explain why. (5) Calculate 7101 Problem 8. Suppose A is an 2x2 matrix with trace equal to 1 and determinant equal to Shou that = 1 Problem 9. Suppose A is an 3 x 3 matrix and tr(A)= 6. tr(A²) 14. and get(A) 6. Find the the characteristic polynomial of A Problem 10. Suppose V is e 20-dimensional linear space over c. and T V - V is a lineas transformation With the characteriatic polynomial fr(x) (At 2)°( (1)(1-1)°(x-2)", and suppose we Anow dim(Im (T 21³ 16 dim/Im (T 1)= 19 dim(Im (T 19 dim(Im (T 21)-) 17 (1) (2) dim(Im (T 27)) dim( Im (T -1)2) dim(Im -21)) (3) tr(T2++T) det(T2 -T) (4) For 21. find all its eigenvalue with its algebraic and geometric multiplicity, and its minimal polynomial 13

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