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MATH 232 Final Exam
August 7th, 2001
Time allowed is 3 hours. There are 19 questions, one per page. Attempt all questions. The total number of marks is 150. A one page sheet of handwritten notes (on both sides) is allowed in the exam. Calculators are not permitted. Put your answers in the space provided. If you need more space, use the back of a page. Please fill in your
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Answer true (T) if the following statements are always true and false (F)
otherwise. You will receive one mark for each correct answer, no mark for no answer and lose one mark for each incorrect answer.
In the following u, v โ Rn^ and A, B, C โ Rnรn.
(a) If ||u|| = 0 then u = 0.
(b) ||u โ v|| โค ||u|| โ ||v||.
(c) ||u + v|| โค ||u|| + ||v||.
(d) AB = BA
(e) A(BC) = (AB)C
(f) If AB = AC then B = C.
(g) If A is singular then the linear system Ax = u has no solution.
(h) If A is invertible then the column vectors of A are linearly inde- pendent.
(i) A linear system with the same number of equations as unknowns has a unique solution.
(j) A linear system with more equations than unknowns has no solu- tion.
Consider the following linear system in x 1 , x 2 , x 3 , x 4.
x 1 + 2x 2 โ 3 x 3 + x 4 = 1 , 3 x 1 + 6x 2 โ 8 x 3 + x 4 = 2.
Write it in the form Ax = b. Solve it by reducing the augmented matrix [A|b] to REDUCED row Echelon form. Express the solutions as a SET of vectors or points in R^4.
Let u = [1, 0 , 1], v = [1, 2 , 3] and w = [0, 1 , 1] be vectors in R^3. Determine whether or not u โ sp(v, w) by setting up a linear system in the form Ax = b and determining whether it is consistent or not. Show your working.
Answer true (T) if the following statements are always true and false (F)
otherwise. You will receive one mark for each correct answer, no mark for no answer, and lose a mark for each incorrect answer.
Let V be a (real) Vector space and let u, v and w be distinct vectors in V.
(a) u + v = v + u. (b) Every subspace of V contains the zero vector of V. (c) If W is a subspace of V and u, v โ W then u โ v โ W. (d) sp(u, v + w) is a subspace of V. (e) The set union of two subspaces of V is a subspace of V. (f) If {u, v} is a basis for V then {u, v, w} cannot be a basis for V. (g) If {u, v, w} is linearly independent then dim(V ) = 3. (h) If dim(V ) = 2 then {u, v, w} is linearly dependent. (i) If dim(V ) = 3 then {u, v, w} is a basis for V. (j) If sp(u, v, w) = V then every vector in V can be expressed as a linear combination of u, v, w.
Let T be a linear transformation from R^2 to R^2. Suppose T ([1, 1]) = [1, โ1] and T ([1, 0]) = [โ 1 , 0]. Determine the standard matrix representation for T , i.e. find the matrix A such that T (x) = Ax. Show your working.
Let B = {u, v, w} where u = x^2 โ 1 , v = x โ 1, and w = x^2 + x. Show that B is a basis for the vector space R[x]/x^3 = sp(1, x, x^2 ).
Let U and V be subspaces of a vector space W. Prove that U โฉ V is a subspace of W.
Calculate the determinant of the following matrix, firstly, by expanding along a row or down a column (your choice), AND secondly, by using Gaussian elimination to reduce A to row Echelon form. Show your working.
A =
Define the eigenvalues and eigenvectors of square matrix A. Give an example of a 2 by 2 matrix A over R which has complex eigenvalues. What are its eigenvalues?
Let A โ Rnรn. Prove that det(A) = 0 if and only if A has a zero eigenvalue.
Let W = sp([1, 1 , 1]). Thus W is a subspace of R^3. Calculate a basis for W โฅ^ the orthogonal complement of W. Find bW the projection of the vector b = [3, 1 , 2] onto W. Find also bW โฅ the projection of b onto W โฅ.
Prove that the projection matrix P is symmetric and idempotent. You may assume P = A(AT^ A)โ^1 AT^ where A is an n by k matrix whose columns form a basis for a subspace W of Rn.
Answer true (T) if the following statements are true and false (F) otherwise. You will receive one mark for each correct answer, no mark for no answer and lose one mark for each incorrect answer.
In the following A =
[
]
(a) A is idempotent. (b) A is symmetric. (c) A is nilpotent. (d) A is invertible. (e) A is row equivalent to I. (f) A has nullspace { 0 }. (g) A has rank 2. (h) A has a zero eigenvalue. (i) A is diagonalizable. (j) A is a projection matrix.
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