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This is the Exam of Linear Algebra which includes Largest Possible Rank, Matrix, Smallest, Possible Dimension, Matrix, Distance, Vector, Linear Transformation, Matrix etc. Key important points are: Enough Work, Vector, Matrix, Find Matrices, Equation, Simplify, Data Points, Least Squares, Best Fits, Values
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Math 205 - Final Exam - April 10, 2006
Instructions: Show enough work to justify your final answers.
, u 1 =
, u 2 =
, and let W =Span{u 1 , u 2 }.
(a) Is y in W?
(b) Find the vector in W that is closest to y.
(c) Find a vector in W ⊥.
(a) Show that A is diagonalizable. That is, find matrices P and D and write the equation involving A, P , and D.
(b) Using your result in part (a), simplify Ak. (Your answer should be a single matrix.)
(a) Is A invertible? Why or why not?
(b) Is A diagonalizable? Why or why not?
(c) Suppose u and v are in the eigenspace of λ = −3. Is it possible that u and v are linearly independent?
(d) Suppose w is in the eigenspace of λ = 2. Calculate A^5 w.
The eigenvalues are 5, 2, and -1, with corresponding eigenvectors
, and
(a) Write out Q(x) in terms of x 1 , x 2 , and x 3.
(b) Find a matrix P such that the change of variables x = P y transforms the quadratic form into one with no cross-product term.
(c) Write the new quadratic form with no cross-product term.
(d) Bonus: Find a vector x such that Q(x) is negative.
(b) Is T a one-to-one transformation? Explain.
(c) Is T an onto transformation? Explain.
