| 1. |
(15%) Consider  . Find u2
, u3 such that {u1, u2, u3}
is an orthogonal set.
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| 2. |
(15%) COnsider  . Find P
such that P-1 AP be a diagonal matrix.
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| 3. |
(15%) Using the property det ( AB )= det (BA) to prove that similar matrices, e.g., A and A', have the same eigenvalues.
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| 4. |
(15%) Consider the linear mapping T : R2 → R2 with T (1,1) = (2,4), T (0,1)=(3,5). Then T (-1,3)= ?
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| 5. |
(15%) Let matrix  . Prove N(A)⊥R(A┬).
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| 6. |
(15%) Consider the quadratic form
where x2+y2+z2=1.
| (a) | Find the maximum value of q(x,y,z). |
| (b) | Also, find (x,y,z) for the maximum q(x,y,z). |
|
| |
| 7. |
(10%) Determine whether or not the following are subspace of R2
| (a) | {(x1,x2)┬│x1+x2=0 |
| (b) | {(x1,x2)┬│x1x2=0} |
| (c) | {(x1,x2)┬│x1=3x2} |
| (d) | {(x1,x2)┬│x1=3x2+1} |
| Explain your answer! |
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