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1 MATH Final Exam (Version ) Solutions July 8, 8 S. F. Ellermeyer Name Instructions. Remember to include all important details of your work. You will not get full credit (or perhaps even any partial credit) if I see gaps in your reasoning. Also, use correct notation and write in complete sentences where appropriate. You may use a calculator on this exam but you may not use any books or notes. There are questions on this exam. You must do numbers,, and. From the remaining questions ( ), you must choose out of from each of the following groups. ircle (in the tables below) the three problems that you want me to grade from each group. I will grade only the three that you circle from each group even if you work on all four. In total, you will be doing problems. hoose three of: hoose three of: 8 9 hoose three of:. (De nitions) Use complete sentences to write the following de nitions. (a) What is a meant by the dimension of a vector space? (b) What is meant by the rank of a matrix? (c) What is meant by the direct sum of two subspaces, H and K, of a vector space V? (d) What does it mean for two subspaces, H and K, of a vector space V to be orthogonal to each other? (e) What does it mean for two subspaces, H and K, of a vector space V to be complementary to each other?

2 . Determine the value (or values) of h such that the matrix h is the augmented matrix of a consistent linear system. Show all of your work. State your conclusion using a complete sentence. Solution: h + h shows that this system is consistent if and only if h =.

3 . Determine whether or not the set of vectors 8 < ; ; : = ; is linearly independent. There is more than one way to do this but you must explain how you make this determination. Solution: Since A = 9 8 the given set of vectors is linearly independent. = I,

4 . Write a system of linear equations that describes the tra c network pictured below. 8 A x x x 8 Find the general solution of this system and then nd the particular solution that corresponds to a ow of x = cars per minute. Solution: The system that describes this tra c network is x + x = 8 x = x + 8 = x + x +. This system has general solution x = x ; x = x, x = free. If x =, then it must be the case that x = and x =.

5 . Suppose that fv ; v ; : : : ; v n g is a set of vectors that spans V n and suppose that T : V n! V n is a linear transformation such that T (v ) = v, T (v ) = v,..., T (v n ) = v n. Prove that T (v) = v for all vectors v V n. In other words, prove that T must be the identity transformation of V n. (Write in complete sentences.) Explanation: Since the set fv ; v ; : : : ; v n g spans V n, then if v is any vector in V n, we know that there exist scalars c ; c ; : : : ; c n such that Since T is linear transformation, then v = c v + c v + + c n v n. T (v) = T (c v + c v + + c n v n ) = c T (v ) + c T (v ) + + c n T (v n ) = c v + c v + + c n v n = v. This shows that T is the identity transformation of V n.

6 . Let A be the matrix A = 9 onstruct matrices and such that A = A but =. (Include all details of how you discover matrices and. Solution: Let Then A = A implies that. b b = c c and = b b c c b + 9b = c + 9c b + 9b = c + 9c b b = c c b b = c c. All solutions of this system must satisfy b = c b + c and b = c b + c. y setting c =, b =, c =, c =, b =, and c = we obtain = and =.. Note that = and A = 9 = 8 and so A = A. A = 9 = 8

7 . Let e, e, and e be the standard basis vectors for V and suppose that T : V! V is a linear transformation such that T (e ) =, T (e ) =, and T (e ) =. (a) Is T one to one? (b) Is T onto V? You must justify your answers to these questions in order to receive credit. Solution: The matrix of this linear transformation is A = The fact that not every column of A is a pivot column tells us that T is not one to one. The fact that every row of A does contain a pivot tells us that T is onto V n..

8 8. Suppose that A and are matrices such that det (A) = and det () =. Use properties of determinants to decide whether each of the following statements is true or false. Explain each of your answers. (No explanation=no credit). (a) det (A) = (True, False) Explanation: det (A) = det (A) det () = () ( ) =. (b) det (A) = (True, False) Explanation: det (ka) = k n det (A) = () =. (c) det T = (True, False) Explanation: det T = det () =. (d) det (A ) = (True, False) Explanation: det (A ) = = det (A) = = =. (e) det (A ) = (True, False) Explanation: det (A ) = (det (A)) = =. 8

9 9. onstruct a geometric gure that illustrates why a line in R that does not pass through the origin is not closed under vector addition. Also provide a written explanation that refers to the gure. Answer: A line, L, that does not pass through the origin is pictured. The points A and are on this line. The position vectors of these points are OA! and O.! It can be seen that! OA + O! = O.! However the point is not on line L. Therefore this line is not closed under vector addition. 9

10 . Suppose that V and W are vector spaces and suppose that T : V! W is a one to one linear transformation. Also suppose that fv ; v ; : : : ; v p g is a linearly independent set of vectors in V. Prove that the set of vectors ft (v ); T (v ); : : : ; T (v p )g is linearly independent (in W ). Hint: Prove the contrapositive. That is, prove that if ft (v ); T (v ); : : : ; T (v p )g is linearly dependent, then fv ; v ; : : : ; v p g must also be linearly dependent. (Remember that we are given that T is one to one. If we were not give that T is one to one, then the statement that we are trying to prove would not even be true.) This question was on Exam (Version ).

11 . Suppose that A is an nn matrix, all of whose entries are integers, and that det (A) =. Explain why all of the entries of A must be integers. This is an essay question. e sure to write in clear and complete sentences. (Hint: A good answer to this question will probably involve mention of cofactors and of the adjugate matrix of A.) Answer: Recall that A ij is the the (n ) (n ) matrix obtained by deleting the ith row and jth column of A. Since all entries of A are integers, then all entries of each matrix A ij are integers. Also, the cofactor, ij is ( ) i+j det (A ij ). It is thus clear (from knowledge of how determinants are computed) that each number ij is an integer. The adjugate of A; denoted by adj (A), is the transpose of the matrix of cofactors of A. Thus each entry of adj (A) is an integer. Finally, since A = adj (A) = adj (A) det (A) (because det (A) = ), we see that each entry of A is an integer.

12 . Let v = ; v = and let H = Span (v ) and K = Span (v ; v ). ; v = (a) Are H and K orthogonal to each other? Explain. (No explanation = no credit.) (b) Is it true that H \ K = f g? Explain. (No explanation = no credit.) (c) Are H and K complementary to each other? Explain. (No explanation = no credit.) Answer: H and K are orthogonal to each other because v T v = [] and v T v = []. (This implies that if u is any vector in H and w is any vector in K, then u T w = [].) The fact that H \ K = f g follows from the fact that H and K are orthogonal to each other. Also, by observing that we see that Span (v ; v ; v ) = V and hence that H K = V. Thus H and K are complementary to each other.,

13 . Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one xed non zero vector? Discuss. (The discussion is what counts here. Write in clear and complete sentences.) Answer: The coe cient matrix of the system being described is of size. We are asking whether or not it is possible for the nullspace of a such a matrix to have dimension. Since a matrix must have at least non pivot columns, then the dimension of the nullspace of such a matrix must have dimension at least. Therefore it is not possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one xed non zero vector.

14 . Find bases for the four fundamental subspaces of the matrix A =. Suggestion: heck your answers by verifying that certain orthogonality conditions (guaranteed by the Fundamental Theorem of Linear Algebra) are present. Solution: Since A =, we see that ol (A) = ; ; A = V. and row (A) = ; ; A. In addition, every vector in Nul (A) has the form x = x x x x x = x x x = x + x. Thus Nul (A) = ; A. The fact that row (A) and Nul (A) are orthogonal to each other follows from the fact that each vector in the set 8 >< >: ; ; 9 >= >;

15 is orthogonal to each vector in the set 8 >< ; >: 9 >=. >; To nd Nul A T, we observe that A T = which tells us that Nul A T = f g. It is clear that ol (A) and Nul A T are orthogonal to each other because is orthogonal to every vector in V.

16 . Decide whether each of the following statements is true or false. (a) If A and are n n matrices, both of which are invertible, then A is invertible. (True, False) (b) If A is an n n matrix and the linear transformation x! Ax is one to one, then the linear transformation x! Ax maps V n onto V n. (True, False) (c) If A is an n n matrix such that det (A) =, then A must either contain a row of zeros or a column of zeros or contain two equal rows or two equal columns. (True, False) (d) Any non empty subset of a vector space is a subspace of that vector space. (True, False) (e) If V is a vector space and S = fv ; v ; : : : ; v p g is a set of vectors in V such that V = Span (S), then dim (V ) p. (True, False) (f) If A is any n n matrix, then row (A) = ol (A). (True, False) (g) If A is any matrix, then row (A) = row (rref (A)). (True, False) (h) If v and v are any two linearly independent vectors in V, then Span (v ) Span (v ) = V. (True, False) (i) If A is a matrix, then Nul A T is a subspace of V. (True, False) (j) If A is any matrix, then ol (A) and row (A) are orthogonal to each other. (True, False)