MATH22 Linear Algebra B Homework Week 4 Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra and its Applications. Pearson, 26. ISBN -52-2873-4. Normally, homework assignments will consist of some odd numbered exercises from the Textbook. The Textbook contains answers to most odd numbered exercises. Be prepared to answer the following oral questions if asked in the supervision class. [.7.2,.22,.33,.35,.37] True or False?. The columns of a matrix A are linearly dependent if the equation Ax = has only the trivial solution. 2. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. 3. The columns of any 4 5 matrix are linearly dependent. 4. If vectors x and y are linearly independent, and if the set { x, y, z } is linearly dependent, then z is in Span{ x, y }. 5. If a set contains fewer vectors then there are entries in the vectors, then the set is linearly independent. 6. If vectors x and y are linearly independent, and z is in Span{ x, y }, then { x, y, z } is linearly dependent. 7. If a set in R n is linearly dependent, then the set contains more vectors that there are entries in each vector. 8. If v,..., v 4 R 4 and then the set v 3 = 2v + v 2, { v, v 2, v 3, v 4, } is linearly dependent. 9. If v and v 2 are in R 4 and v 2 is not a scalar multiple of v, then the set { v, v 2 } is linearly independent.. If v,..., v 4 R 4 and { v, v 2, v 3 } is linearly dependent then { v, v 2, v 3 v 4 } is also linearly dependent. 2. [.8.2 and.8.22] True or False?. A linear transformation is a special type of function. 2. If A is a 3 5 matrix and T is a transformation defined by T (x) = Ax, then the domain of T is R 3. 3. If A is an m n matrix, then the range of transformation is R m. 4. Every linear transformation is a matrix transformation. 5. A transformation T is linear if and only if T (c v + c 2 v 2 ) = c T (v ) + c 2 T (v 2 ) for all v and v 2 in the domain of T and for all scalars c and c 2. 6. Every matrix transformation is a linear transformation. 7. The codomain of the transformation is the set of all linear combinations of the columns of A. 8. A linear transformation preserves the operation of a vector addition and scalar multiplication. 3. [.9.23 and.9.24] True or False?. A linear transformation T : R n R m is completely determined by its effects on the columns of the n n identity matrix.
MATH22 Linear Algebra B Homework Week 4 2 2. When two linear transformations are preformed one after another, the combined effect may not always be a linear transformation. 3. A mapping T : R n R m in onto R m if every vector x in R n maps onto some vector in R m. 4. If A is a 3 2 matrix, then the transformation cannot be one-to-one. 5. Not every linear transformation from R n to R m is a matrix transformation. 6. The columns of the standard matrix for a linear transformation from R n to R m are the images of the columns of the n n identity matrix. 7. A mapping T : R n R m is one-to-one if each vector in R n is maps onto a unique vector in R m. 8. If A is a 3 2 matrix, then the transformation cannot map R 2 onto R 3. 4 5. These problem numbers are reserved for extra questions for exam revision Solve the following exercises (but do not submit them for marking) 6. [.4.5,.7] Write the matrix equation as a vector equation, or vice versa. x [ 5 8 4 2 7 3 5 5 [ ] 8 3 = ; 6 2 ] 4 5 7 6 7 + x 3 2 5 + x 8 3 = 8. 4 2 7 7. [.4.] Write the augmented matrix for the linear system that corresponds to the matrix equation Ax = b. The solve the system and write the solution as a vector. 2 4 2 A = 5, b = 2. 2 4 3 9 8. [.4.5] Let A = Show that the equation [ ] [ ] 2 b, b =. 6 3 b 2 Ax = b does not necessarily have solution for every possible b, and describe those vectors b for which Ax = b does have a solution. 9. [.4.7,.9] Let 3 3 A = 4 2 8. 2 3 How many rows of A contain a pivot position? Does the equation Ax = b have a solution for each b R 4? Can each vector in R 4 be written as a linear combination of the columns of the matrix A? Do the columns of A span R 4?. [.5.] Determine whether the system has a nontrivial solution. Try to use as few row operations as possible. 2x 5x 2 + 8x 3 = 2x 7x 2 + x 3 = 4x + 2x 2 + 7x 3 = And the same question for his system: 3x + 5x 2 4x 3 = 6x + 3x 2 2x 3 =. [.5.5] Find the solution set of the given
MATH22 Linear Algebra B Homework Week 4 3 homogeneous system in parametric vector form: x + 3x 2 + x 3 = 4x 9x 2 + 2x 3 = 3x 2 6x 3 = 2. [.5.5] Find the solution set in parametric vector form: x + 3x 2 + x 3 = 4x 9x 2 + 2x 3 = 3x 2 6x 3 = 3 3. [.7.] Are the vectors linearly independent? 5 7 9, 2, 4 6 8 4. [.7.5] Do the columns of the matrix form a linearly independent set? 8 5 3 7 4 5 4 3 2 5. Do the columns of the matrix form a linearly independent set? 2 3 2 3 2 2 2 3 3 3 6. [.7.9] For which values of h is v 3 is in Span{ v, v 2 }? 3 5 v = 3, v 2 = 9, v 3 = 7 2 6 h 7. [.7.] For which values of h are the vectors linearly dependent?, 4 3 5, 7 5 h 8. [.7.5,.7,.9] Determine by inspection whether the vectors are linearly independent. [ ] [ ] [ ] [ ] 5 2 (a),,, ; 8 3 7 (b) (c) 3 5,, 6 5 ; 4 8 2 2, 3. 4 9. [.7.23] Describe the possible echelon forms of a 3 3 matrix A with linearly independent columns. 2. [.7.27] How many pivot columns must a 7 5 matrix have if its columns are linearly independent? Why? 2. [.7.29] Construct 3 2 matrices A and B such that Ax = has only the trivial solution and Bx = has a non-trivial solution. 22. Given 2 A = 3 4 4 5, 6 7 observe that the third column is the sum of the first two. Without doing row transformation, find a nontrivial solution of Ax =. Submit for marking solutions for Problems 23* 26* 23.* Let b A = 2 3, b = b 2. 3 4 b 3 (a) Show that the equation Ax = b does not have a solution for all b (but can have solutions for some b). (b) Find all vectors b = b b 2 b 3 such that the equation Ax = b has a solution and, in addition, satisfy b + b 2 + b 3 =.
MATH22 Linear Algebra B Homework Week 4 4 24.* Determine if the vectors are linearly dependent. 25.* Given 2, 5 8, 3 4 4 6 A = 7 5 3, 9 3 3 observe that the first column plus twice the second column equals the third column. Find a nontrivial solution of Ax =. 26.* (a) Fill in the blank in the following statement: if A is an m n matrix then the columns of A are linearly independent if and only if A has... pivot columns. (b) Explain why the statement in (a) is true. Answers to oral questions. [.7.2,.22,.33,.35,.37]. Answer: False; the correct statement is The columns of a matrix A are linearly dependent if the equation Ax = has a nontrivial solution. 2. Answer: False. A counterexample (one of many possible) is { [ [ [ } v =, v ] 2 =, v ] 3 = ; ] the vector v 3 is not a linear combination of v and v 2. 3. The Answer: True. 4. Answer: True. 5. Answer: False. A simple counterexample is, 2 2. 2 6. Answer: True. Since z Span{ x, y }, z = ax + by for some constant a and b, which gives us the linear dependency of x, y, and z: ax + by z =. 7. Answer: False. See 5. 8. Answer: True, since 2v + v 2 v 3 + v 4 =. 9. Answer: False. A counterexample is v = and v 2 =.. Answer: True. Since { v, v 2, v 3 } is linearly dependent, c v + c 2 v 2 + c 3 v 3 = for some constants c, c 2, c 3, one of which is nonzero. But then c v + c 2 v 2 + c 3 v 3 + v 4 = tells us that { v, v 2, v 3, v 4 } is also linearly dependent. 2. [.8.2 and.8.22] True or False?. Answer: True. 2. Answer: False. 3. Answer: False: take for A the zero matrix. 4. Answer: True. 5. Answer: True. 6. Answer: True. 7. Answer: False. It is the range of the transformation which is the set of all linear combinations of the columns of A. 8. Answer: True. 3. [.9.23 and.9.24]. Answer: True. 2. Answer: False. 3. Answer: False.
MATH22 Linear Algebra B Homework Week 4 5 4. Answer: False. A counterexample is A = ; indeed the map [ ] x x 2 is one-to-one. 5. Answer: False. [ ] x = x x x 2 2 6. Answer: True. 7. Answer: True. 8. Answer: True. 4 5. These problem numbers are reserved for extra questions for exam revision Answers to non-starred exercises 6. [.4.5,.7] Answer: [ ] [ ] [ ] [ ] 5 8 4 5 + 3 2 = 2 7 3 5 4 5 7 3 8 x 6 7 5 x 2 = 8 x 4 2 3 7 7. [.4.] Answer: 2 4 2 5 2, 2 4 3 9 x x 2 = 3. x 3 [ ] 8 ; 6 8. [.4.5] Answer: The equation Ax = b is not consistent when 3b + b 2. The set of b for which the equation is consistent is made of all vectors of the form [ ] b = b for all b R. 3 9. [.4.7,.9] Answer: Only three rows contain a pivot position, therefore answer to other three questions are all negative.. [.5.] Answer: The systems have a nontrivial solution because each of them contain a free variable, x 3.. [.5.5] Answer: 2. [.5.5] Answer: x x 2 = x 3 5 2 x 3 x x 2 = 2 + x 3 5 2 x 3 3. [.7.] Answer: Yes. 4. [.7.5] Answer: Yes. 5. Answer: No, they do not: there are too many of them. 6. [.7.9] Answer: No h (look at the first two rows in the column vectors). 7. [.7.] Answer: h = 6. 8. [.7.5,.7,.9] Answer: (a) No; (b) no; (c) yes. 9. [.7.23] Answer: where denotes arbitrary nonzero number and any number. 2. [.7.27] Answer: All five columns of the 7 5 matrix A must be pivot columns. Otherwise Ax = would have a free variable, in which case the columns of A would be linearly dependent. 2. [.7.29] Answer: A: Any 3 2 matrix with nonzero columns such that neither column is a multiple of the other. B: Any 3 2 matrix with one column a multiple of another. 22. Answer: x = P.T.O.
MATH22 Linear Algebra B Homework Week 4 6 Answers to starred questions 23.* (a) The last row of the reduced echelon form of the augmented matrix of the system is [ b b 2 + b 3 ], hence the solution exists only when b + b 2 b 3 =. (b) They are found from the system of equations b + b 2 b 3 = b + b 2 + b 3 = which has solutions b b 2 = b. b 3 24.* The vectors are independent, just swap rows and 3 in the matrix made of vectors, the result will be in echelon form where this is obvious. 25.* x x 2 = 2. 26.* (a) n. x 3 (b) Non-pivoted columns are linear combinations of pivoted columns. Indeed that follows from the procedure for solving the systems of homogeneous linear equations. Indeed assume for simplicity that columns a,..., a r of the matrix A are pivoted and a r+..., a n are non-pivoted. Then in the system where Ax =, x x =. x n the variables x r+,..., x n are free. Let now a k be a non-pivoted column. Set x k = and set the rest of free variables being equal. Then we can find pivoted variables x,..., x r, for example, x i = c i, and we have or c x + + c r a r + a k = a k = c x c r a r is a linear combination of pivoted columns. It is worth emphasising again that this is nothing more but procedure for solving systems of homogeneous linear equations, only expressed in a slightly different terminology!