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Solutions to the final exam of math 308, held on dec. 8th, 2008. The exam covers topics in linear algebra and matrix computations, including solving linear systems of equations, computing eigenvalues and eigenvectors, determining the rank and nullity of matrices, and applying gram-schmidt orthogonalization.
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Math 308, Dec. 8th 2008
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The exam is 1h 50 minutes long. No textbook allowed. Students are allowed to bring two sheets of personal notes (two sided), and a scientific calculator.
Solve the following linear system of equations:
x 1 − x 2 + x 3 = 0 2 x 1 − x 2 = 0 − 2 x 1 + 3 x 2 − 3 x 3 = 10
Let A =
Compute the characteristic polynomial of A and all its eigenvalues. [Hint: one of them is 3 and another is 2 , check your computations if you did not find it]
Compute the eigenspace for the eigenvalue λ = 3.
Let V = R(A), (the range of the matrix A), where A =
Determine a basis for V and its dimension.
Determine the rank and the nullity of A (without computing a basis of the nullspace).
Let w 1 =
(^) , w 2 =
(^) , w 3 =
Show that w 1 , w 2 , w 3 form a basis of R^3.
Apply Gram Schmidt orthogonalization to find an orthogonal basis of R^3.
We will now prove that they are orthogonal
Show that the scalar product of Au 1 and u 2 is equal to the scalar product of u 1 and Au 2 [Hint: recall that (Au)T^ = uT^ AT^ .]
Deduce from question 3) that (λ 1 − λ 2 )uT 1 u 2 = 0
Conclude that u 1 and u 2 are orthogonal.
Consider the following data table: t -1 0 2 y 1 4 4
An n × n matrix is called skew symmetric if
AT^ = −A.
Show that if A is skew symmetric, then det(A) = (−1)n^ det(A).
Argue that if n is odd, then an n × n skew symmetric matrix is singular.
