KEY. Math 343 Midterm 2 Instructor: Scott Glasgow Sections: 1, 6 and 8 Dates: October 26 th and 27 th, 2005

Transcription

1 KEY Math 4 Midterm Instructor: Scott Glasgow Sections:, 6 and 8 Dates: October 6 th and 7 th, 005 Instructions: You will be doced points for irrelevant developments as well as patently wrong ones So organize your efforts on two sets of papers, one set called scratch containing whatever meanderings you needed to get things clear in your mind, and which set I do not ever want to see, the other set on which you put your name and section number containing only relevant calculations and discussions leading directly to (hopefully) correct answers, and which you hand in to the testing center staff (so that I may see and grade them) Related to this idea is the fact that you will also be doced for handwriting and organization that is so poor that either I cannot determine your development, or I find it painful to determine it In summary, your wor that I ultimately see and grade should be a (logical) wor of art

2 4 Midterm Fall 005 Demonstrate the validity of the Cauchy-Schwarz inequality for u (, 4) and v (5,) In particular, calculate u v u v 7 pts : The Cauchy-Schwarz inequality is (for now) that for any u and v in a Euclidean space n, Here we have that u u v u v u u + (, 4) (, 4) 4 5 5, and indeed v v v + (5,) (5,) 5, u v (,4) (5,) u v Specifically we find that u v u v 65 6 What does the Wronsian tell you about the following sets of vectors in C ( )? a) {, sin, cos }, x, x + 4, c) { x,x+,9x 5} 5 pts x x, b) { } We have a) sinx cosx cos x sin x cos x sin x W{,sin x, cos x} 0 cos x sin x sin x cos x sin x cos x 0 sinx cosx ( x x) cos + sin 0 for some (all) value(s) of x, whence the set is linearly independent, b)

3 { } x x + 4 W, x, x x for some (all) value(s) of x, whence the set is linearly independent, and c) { } x x+ 9x5 W x,x+,9x for all values of x, whence use of the Wronsian is inconclusive (but it is clear that the set is linearly dependent directly from the definition since x 5 x+ 9x 5 0 for every value of x ) ( ) ( ) The vector b (, 9, ) is in the column space of A So write b as a specific linear combination of A s columns by solving A x b 5 pts As indicated, b is in ColA iff there is an x such that A x b So we attempt to solve the latter for x, here row reducing the augmented matrix A b Thus A

4 b 9 ( ) + + { } S is a basis for, find 4 Given that the set (,,0 ),(,, ),(,0,) a) the coordinate vector of v (,,) with respect to S, b) a vector v in whose coordinate vector with respect to S is (,,) 5 and 5 pts a) By definition, the coordinate vector ( ) (,, ) that whence v c S c c that we see has the property ( ) ( ) ( ) v (,,) c,,0 + c,, + c,0,, c 0 c, 0 c the augmented matrix for which being , v c, c, c,,4 S so that ( ) ( ) ( ) b) If ( v ) (,, ) then S v (,,0) + (,,) + (,0,) ( + +, + + 0, 0+ + ) (,0,4 ) 5 Find the coordinate vector ( ) of p p + 5x + 9x P with respect to the basis S {, 9 7, 8 } S x+ x + x+ x x+ x for P 5 pts

5 By definition, the coordinate vector ( ) (,, ) Since S {, x, x } p c S c c that we see has the property that p + 5x + 9x c x+ x + c + 9x+ 7x + c 8 x+ x ( ) ( ) ( ) is a basis for, the vector equation above is the same as the scalar P equations c+ c 8c, c+ 9c c 5,c+ 7c + c 9, or the augmented matrix , so that ( p) ( c, c, c) (,,) S 6 For each of parts a), b), and c) below, either determine a nontrivial linear relationship among the set of vectors in, or conclude that the set is a basis of by showing the set is linearly independent: a) v (, 4,), v ( 5, 7,8), and v ( 6,, 4), b) v (,7, 4), v ( 6, 8,), and v ( ) v ( ) v ( ), and ( ) 6,5,8 8 pts, and c) 6, 5, 7,, 4, v 5,9, a) Linear relationships among vectors can be detected by row reducing matrices whose columns are those vectors:

6 v v v : w w w So since it is clear that w 4w+ w, it follows that 4 + v v v b) 6 6 v v v : w w w So since it is clear that w w+ w, it follows that v v+ v c) v v v : w w w So since it is clear then that,, and w are independent and form a basis for, it w w follows that,, and v are independent and form a basis for v v 7 Let T : be linear, and let 0 T, and T 4 4 Find T x x 8, and, if possible, find such that T Also, does there exist 0 y y 8 u, v, u v, such that Tu Tv? Finally, can you find a u, u 0, such that Tu λu u? If so, give such a scalar λ ( )

7 0 pts Note that by linearity T T T T T T T 8, which answers the first question (and the last) Note also that T T T T, 8 which answers the second question Note the next question regards whether T is one-toone, which holds for a linear mapping precisely when T x 0 x 0, ie when T, and so[ T ], is invertible (since T maps a space into itself) To answer that question, construct [ T ] by finding 0 T : Note that 6 0 T T T T, so that now, by theorem, [ T] Te Te T T 0 0 0, T (and so T ) is invertible, giving no distinct vectors from which it is obvious that [ ] u, v such that T u Tv From this form it is also obvious that another eigenvalue (besides ) is 8, which is the desired λ, and which gives T u 8u for all u Span, 0 {( )}

8 > (the set of positive numbers written underscored), S, and let 8 Let V { x x 0} vector addition " + " and scalar multiplication " " be defined through for x, y V, x" + " y: xy for S, x V, " " x: x (Here xy denotes an underscored number, the number being xy meaning x times y x denotes an underscored number, the number being and x meaning x raised to the th power) Determine whether the two sets together with the two binary operations constitute a vector space, by demonstrating each satisfied axiom, as well as demonstrating each unsatisfied axiom (You must demonstrate satisfaction or failure of each of the ten axioms) 0 pts If x, y V, then xy>, 0, so that xy > 0, yielding x" + " y: xy V as required For x, y V, x" + " y: xy yx : y" + " x, as required For x, yz, V, x" + "( y" + " z) : x" + " yz: x( yz) ( xy) z : xy" + " z ( x y) : " + " " + " z, as required 4 Note that V has the property that, for any x V, " + " x x x x x" + ", so that V is the zero vector, or additive identity, as required 5 Note that since x 0 x > >0, for each x V there exists x V, having the property that + + x " " x x x x x x" " x negative (or additive inverse) to be x : x, as required, so that we can tae the 6 Since, x > 0 x >0, we have that, for S, x V, "" x: x V, as required 7 We have ( ) ( ) "" x" + " y : "" xy : xy x y : x " + " y : " " x" + " " " y", as required + m m m 8 We have ( + m)" " x: x x x : x " + " x : " " x+ m" " x, as required ""( m""): x ""( x ): x x :( m)"" x, as required m m ( m) 9 We have ( ) 0 We have " " x : x x, as required 9 Using the axioms recalled in problem 8, prove that, in any vector space, 0 u 0for any u Here be sure to use that a) the 0 here is the usual 0, namely the additive identity in the scalars (generally called the field ), but that b) 0 is the additive identity in the

9 vectors You will be given points for each relevant use of an axiom (and doced for each irrelevant use) (And you do not have to remember the axioms in the same order the boo lists them, but you do have to be clear which one you are using) 6 pts Since 0 is the additive identity in the scalars, we have that , and then that for any, 0u 0+ 0 u But then by axiom 8 we have u ( ) ( ) 0u 0+ 0 u 0u+ 0u (0) Now by axiom 5 there exists (0) gives 0u such that 0u+ ( 0u) 0 Adding such to each side of ( ) ( ) ( ) 0u+ 0u 0u+ 0u + 0u, or 0 ( 0u+ 0u) + ( 0u ) (0) Using axiom, the latter becomes 0 0u+ ( 0u+ ( 0u )) or, using axiom 5 again, 0 0u+ 0 (0) Using axiom 4, the latter is 0 0u+ 0 0u, ie 0 u 0 0 If 0 0 cosθ 0 sinθ cosθ sinθ 0 Rx( θ) 0 cosθ sin θ, Ry( θ ) 0 0, Rz( θ ) sinθ cosθ 0 0 sinθ cosθ sinθ 0 cosθ 0 0 are the standard matrices for counter-cloc-wise rotations of θ radians in about the indicated coordinate axes, then find the standard matrix T for the linear operators giving a) a counter-cloc-wise rotation of π / radians about the z -axis, followed by a countercloc-wise rotation of π / radians about the y -axis, followed by a counter-cloc-wise rotation of π / radians about the x -axis, and b) a reflection about the y z plane, followed by a dilation of / (Recall that one need not actually use any of the matrices above, but rather all one has to do is find out what the linear mappings do to unit vectors along the (standard) coordinate axes) [ ]

10 9 and 5 pts a) By theorem 0 0 [ T] Te Te Te T 0 T T Now under these three mappings, first gets sent to e by the first rotation, and then e e e does not change under the second, and finally gets sent toe by the third Thus T e e On the other hand, e first gets sent to e by the first rotation, and then e gets mapped to eunder the second, and finally e gets sent to e by the third Thus T e Lastly is unchanged by the first rotation, and then e gets mapped to e e e under the second, and finally e is unchanged by the third Thus T e e, and we now have that [ T] T T T e e e e e e b) Reflecting about the y z plane sends e to e, and leaves e and eunchanged Thus, since the subsequent dilation is rather trivial, we have [ T] T T T e e e e e e