LINEAR ALGEBRA : QUIZ 2 (9:10am-10:50am, May 24, 2006)
Use of all automatic computing machines including calculator is prohibited.
1.Let A be the 4x4 matrix given below.
╭ 2 -2 1 1╮
A= │ -2 2 1 1│
│ -1 -1 3 2│
╰ 1 -1 2 3╯
(a)(8%)Find the characteristic polynomial of A.
(b)(4%)What are the eigen values of A and their multuplicity?
(c)(8%)For each eigen value, find a basis for the corresponding eigenspace.
(d)(4%)Show the A is diagonalizable. And find a basis B草寫 for R^4 consisting
of the eigenvectors of A.
(e)(4%)Find P and D (diagonal) such that A=PDP^-1.
(f)(4%)Can we find a B matrix such that B^2 = A? If we can, what is B?
(g)(3%)Let T be a linear operator on R^4 with standard matrix A. Find [T]B草寫
where B草寫 is difined in (d).
2.Let u1, u2, and u3 be the first 3 columns of the matrix A in Problem 1
and let V = Span{ u1, u2, u3 }.
(a)(10%)Find an orthonormal basis for V.
(b)(5%)Find a basis for V^垂直
(c)(5%)Let v = [ 1 1 1 1 ]^T(轉置). Find the vector in V that is closest to v.
What is the distance from v to V?
(d)(5%)Find the unique vector w in V and the unique vector z in V^垂直
such that v = w + z.
3.
(a)(5%)Show that if A is diagonalizable, that A^2 is also diagonalizable.
(b)(5%)If A^2 is diagonalizable, is A diagonalizable? (Justify your answer.)
4.(10%)Let A be a diagonalizable n x n matrix. Prove that if
f(t) = (an)*t^n + (an-1)*t^(n-1) + ... + (a1)*t + (a0)*t^0
is the characteristic polynomial of A, then f(A) = 0大零.
5.(10%)Prove that if V and W are subspaces of R^n such that V
is contained in W, then W^垂直 is contained in V^垂直.
6.Let u and v be vectors in R^n such that v dot u = 1. Let the n x n matrix
A = u v^T.
(a)(5%)Find all eigenvalues of A.
(b)(5%)Describe the eigenspaces corresponding to the eigenvalues
you found in (a).
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