Week #4: Midterm Review April 5, NAMES: TARDIS : http://math.ucsb.edu/ kgracekennedy/spring 4A.html Week : Introduction to Systems of Linear Equations Problem.. What row operations are allowed and why?... Problem.. Solve the system of equations two ways, side by side: by adding or subtracting the equations, and by row reducing the augmented matrix. x + y + z = 4 x + z = 6 y + z = Problem.. Suppose we have a linear system. What happens to the solution set if we multiply one of the equations by? Find one example where this doesn t change the solution set, and one example where it does.
Midterm Review Problem.4. The following augmented matrix of a linear system has been reduced by row operations to the form shown. Give a vector description of the set of ALL solutions. If there are infinitely many, writing the solutions as a parameter family of solutions (i.e. linear combination of vectors) and using span notation. 5 7 4. Problem.5. Give a vector description of the set of ALL solutions. If there are infinitely many, writing the solutions as a parameter family of solutions (i.e. linear combination of vectors) and using span notation. x 4x 4 = x +x = x +4x 4 = x +x +4x = 5 Problem.6. Give a vector description of the set of ALL solutions. If there are infinitely many, writing the solutions as a parameter family of solutions (i.e. linear combination of vectors) and using span notation. x + x + x + 4x 4 = x + 4x + x + 8x 4 = of 4
Midterm Review Problem.7. Find the solution set of the system corresponding to [ ] 5 5 in the form x x = + s x Problem.8 (.5.6). Write the solution to the system in parametric vector form (i.e. as x x x x + x x = x + x x = x + x = = + s + t + I ll let you determine how many variables you need to use). Problem.9. Consider the augmented matrix A = 4. 4 Does this correspond to a consistent system of equations? If so, does this system have a unique solution? Problem.. Does the augmented matrix 4 have zero, one, or infinitely many solutions? of 4
Midterm Review Problem.. Determine the value(s) of h such that [ h 4 6 8 is the augmented matrix of a consistent linear system. Justify your answer. ] Do the same with [ 4 h 6 ]. Problem.. Find an equation involving g, h, and k which makes this augmented matrix correspond to a consistent system: 4 7 g 5 h. 5 9 k 4 of 4
Midterm Review Problem.. Give an example of an augmented matrix that... or explain why it is impossible to do so. In each case, write the system of linear equations that the augmented matrix represents. How many unknowns are there in each case? has a row of zeros and is inconsistent Augmented Matrix: represents this system of linear equations with solutions: with unknowns. has a row of zeros and infinitely many solutions Augmented Matrix: represents this system of linear equations with solutions: with unknowns. has a row of zeros and has a unique solution Augmented Matrix: represents this system of linear equations with solutions: with unknowns. has a column of zeros and is inconsistent Augmented Matrix: represents this system of linear equations with solutions: with unknowns. has a column of zeros and infinitely many solutions Augmented Matrix: represents this system of linear equations with solutions: with unknowns. has a column of zeros and has a unique solution Augmented Matrix: represents this system of linear equations with solutions: with unknowns. 5 of 4
Midterm Review Week : Span and More on Systems Problem.. Consider the following system: x y + z = x z = Write this as a matrix equation AND as an equation involving vectors. Why are these three representations of the same solution set? Problem. (..). Determine if b is a linear combination of a, a, and a : a = a = a = 6 7 b = 5 5 9 How does this relate to span? Problem.. Is the vector b in Span(a, a, a ) if a =, a = 5, a = 6 and b =? 8 6 Do a, a, and a span R? What about a, a, a and b? Problem.4. Let 4 8 v = v = 5 and w =. 5 8 9 Determine if w is in Span{ v, v }. Does any combination of these vectors span R? How does this relate to the Magic Carpet Ride question? Problem.5. Describe Span. Is it R? Problem.6. Can the span of a single vector ever be R? Can the span of two vectors ever be R? If m < n then can m vectors ever span R n? (Hint: Try to use language from the magic carpet ride project.) 6 of 4
Midterm Review [ ] Problem.7. Let A =. Does A x = b have a solution regardless of what I choose for b? What does that tell you about the span of the columns of A? Problem.8. Let A = 4 5 What does that tell you about the span of the columns of A?. Does A x = b have a solution regardless of what I choose for b? Problem.9 (..). Construct a matrix A, with nonzero entries, and a vector b in R such that b is not in the set spanned by the columns of A. [ ] [ ] b Problem. (.4.5). Let A = and b =. Show that the equation Ax = b does not 9 b have a solution for all possible b, and describe the set of b for which Ax = b does have a solution. How does this relate to span? 7 of 4
Midterm Review Problem. (.4.6). Let u = 7, v =, and w = 5. It can be shown that u v w =. 5 Use this fact (and no row operations) to find x and x that satisfy the equation 7 [ ] 5 x = x 5 How does this relate to linear independence and span? Problem.. In 4 5 A = 5 note that one column is the sum of the other two. Find three different solutions to Ax =. (How does this relate to span?) Note that you do not need to use row operations for this problem. See.4.6 for inspiration. Problem.. For each of the following collections of vectors, find all subcollections of minimal size with the same span. (a) Do these vectors span R? (b) 4 4 Do these vectors span R? 6 (c) Do these vectors span R4? 8 of 4
Midterm Review Week : Linear Independence and Linear Transformations. Linear Independence [ ] Problem.. Let A =. Does A x = have a nontrivial solution? What does that tell you about column vectors of A? Problem.. Let A = 4 5. Does A x = have a nontrivial solution? What does that tell you about the column vectors of A? Problem.. Let A = 4 5. Does A x = have a nontrivial solution? What does that tell you about the column vectors of A? Problem.4. Suppose I gave you a random n m matrix A, and I gave you its RREF. Suppose I ask you the question above (i.e. does the homogeneous system, A x =, have a nontrivial solution? ). How can you immediately know the answer? 9 of 4
Midterm Review Problem.5. Find the values of h that make the following linearly independent: 6 9 6 4 h Problem.6 (.5.). Construct a x nonzero matrix A such that the vector Ax =. What does this say about the columns of A? is a solution of Problem.7 (.5.6). Given A = 6 4 8 row reduction). What does this say about the columns of A?, find one nontrivial solution of Ax = by inspection (not Problem.8 (.5.8,.5.). For each part, determine if Ax = has a nontrivial solution, and also determine if Ax = b has at least one solution for every possible b. If A is a x matrix with three pivot positions? If A is a x5 matrix with two pivot positions? of 4
Midterm Review Problem.9. Are these vectors linearlly independent? 7 Problem.. Find both a linearly independent and a linearly dependent set of vectors that satisfy the following criteria: (a) A set of vectors in R (b) A set of vectors in R (c) A set of vectors in R (d) A set of vectors in R (e) A set of 4 vectors in R Problem.. Can a set of three vectors in R be linearly independent? If m > n then can m vectors in R n be linearly independent? (Hint: Try to use your answer to the previous Problems.4 or..) Problem.. Which of the following sets of vectors is linearly independent? (a) 7 4 (b) 8 4 4 (c) 4 7 4 (d) 8 8 99 4 4 66 of 4
Midterm Review Problem.. For each of the following collections of vectors, find all subcollections of maximal size that are linearly independent. Do they have the same span as the original set? (a) (b) 4 4 6 (c) (d) Problem.4. Show that a subset of any linearly independent set of vectors is linearly independent. Problem.5. Suppose that { v, v,..., v n } is linearly dependent in R p. Show that one of these vectors is a linear combination of the others. of 4
Midterm Review. Linear Transformations Problem.6 (Test Question Last Quarter). Let T : R R be a linear transformation so that [ [ [ [ ] T ( ) =, T ( ) =. ] 5] ] 6 Then [ 4 T ( ) = 4]? [ ] [ ] [ ] [ ] Problem.7 (.8.9). Let e =, e =, y =, and y 5 =, and let T : R 6 R [ ] [ ] 5 x be a linear transformation that maps e into y and e into y. Find the images of and. x Problem.8 (.9.). Find the standard matrix of T if T : R R 4, T (e ) = (,,, ) and T (e ) = ( 5,,, ), where e = (, ) and e = (, ). ([ ]) Problem.9. Let T be a linear transformation so that T = ([ and T 5 Find T ([ 4 6 ]) ([ and T ]). ]) =. of 4
Midterm Review Problem. (.8.). Let A = T (x) = Ax. Find T (u) and T (v)., u = 6 9, v = a b c. Define T : R R by Problem. (Test Question Last Quarter). There is a secret matrix C for which C = e C = e Find a D so that CD = [ ] and solve for the two vectors shown: D = C = [ ] 4, C = [ ] 4 5 Problem. (.9.8). T : R R first performs a horizontal shear that transforms e into e + e (leaving e unchanged) and then reflects points through the line x = x. Find the standard matrix of T. Problem.. Suppose T : R n R m is a linear transformation. If the statement is true, prove it. If the statement is false, give a counterexample. (a) True or False: If {v,, v k } R n is linearly dependent, then {T (v ),, T (v k )} is linearly dependent. (b) True or False: If {v,, v k } R n is linearly independent, then {T (v ),, T (v k )} is linearly independent 4 of 4