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This is the Exam of Matrix Algebra which includes Unique Solution, System, Solutions, Infinitely Many Solutions, General Solution, Parametric Vector, Determinants, Matrices, Traffic Flow etc. Key important points are: Linear System, Values, Solution, System, Map, Rotates Points, Oriented, Formula, Linear Transformation, Standard Matrix
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The University of British Columbia Final Examination - April 20, 2007 Mathematics 221 Sections 201, 202, 203 Instructors: Dr. Macasieb, Dr. Tsai, and Dr. Liu
Closed book examination Time: 2.5 hours
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Total 100
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x + 3y − 2 z + 2w = 1 y + z − 2 w = 2 x + 2y − 2 z + aw = 0 2 x + 8y − z + w = b
For which values of a and b, if any, does the system have: (Justify you answers!!) (i) No solution? (ii) Exactly one solution? (iii) Exactly two solutions? (iv) More than two solutions?
0 1 k
(^) invertible? When it is invertible,
find its inverse.
b + 2c − d 2 b + 4c − d d −b − 2 c + d
b, c, d real
(i) Find a matrix A such that Col A = W.
(ii) Find a basis for W.
(iii) If B =
0 2 0 k 1 1 1 3
and dim (Row^ B) = 2, find the value of the constant^ k.
x 1 1 1 1 1 x 1 1 1 1 1 x 1 1 1 1 1 x 1 1 1 1 1 x
. Find all values of x such that A is not invertible.
(i) The set B = {1 + t, 1 + t^2 , t + t^2 } is a basis for P 2. Find the coordinate vector [2+t−t^2 ]B.
(ii) The set C = {1 + t^2 , t + t^2 , 1 + t} is also a basis for P 2. Find ~p(t) in P 2 such that ~p(1) = 1 and [~p(t)]B = [~p(t)]C.
w ~ 1 =
(^) , ~w 2 =
(^) , ~w 3 =
(^) , ~y =
Let W = Span{ w~ 1 , ~w 2 , ~w 3 }. (i) Determine the dimension of W and find a basis for W. (ii) Find an orthogonal basis for W , and the orthogonal projection of ~y onto W. (iii) What is the shortest distance from ~y to W?
(i) Verify that M has eigenvalues 0 and 3, and find the corresponding eigenspaces. (ii) What is the rank of M? (iii) Is M diagonalizable? Is there an orthogonal set of eigenvectors of M that forms a basis of R^3? Justify your answers.