Alg1231W2T3 - Examples and counterexamples

Wednesday, 25 July 2018

9:10 PM

Machine generated alternative text: Here are the essential features that make R into a vector space [select all that apply]: We have a well-defined notion of addition of vectors We have a well-defined notion of multiplication of vectors We have a well-defined notion of division of vectors We have a well-defined notion of multiplying a vector by a scalar These operations satisfy a particular list of properties The basic list of required properties includes [select all that apply]: If u and v are vectors then u + v = v u Cl If u and v are vectors then u x v = v x u If u, v and w are vectors then (u + v) + w = u + (v + w) Cl If u, v and w are vectors then (u — v) — w = u — (v — w) Cl There is a special vector 1 with the property that 1 x v v for any vector v There is a special vector O with the property that O + v v 0 = v for any vector v If and are scalars and v is a vector then ( Ag)v If and are scalars and v is a vector then (A + = + Note: this is not a comprehensive list, for that see page 4-6 of your yellow notes In fact all of these statements apply not just to making R into a vector space, they also apply to: [select all that apply] the vectors v — the matrices A y , making R into a vector space all (112 making M22 into a vector space (121 (122 41 the polynomials p(x) ao + al x + a,-2X , making P2 (IR) into a vector space.

 

Machine generated alternative text: A subset S of IR2 can often be visualised geometrically A subset S of IR2 is a subspace if it is also a vector space on its own; using the same operations of vector addition and scalar multiplication as in R . Most subsets of R are not subspaces. Subspaces are very special and easy to identify geometrically Match up the following sets with the appropriate diagrams —1 T R s v w velR velR veR y y y with with with with with -3 y 2m -2 and 2 2 2 3 -3 -2 2 3

 

Machine generated alternative text: Which of the subsets above is also a subspace of R ?

 

Machine generated alternative text: Consider the set R Which of the following vector space properties hold for R [select all that apply] 41 1. Closure under Addition. If u, v e R, then u + v e R velR with y 41 2. Associative Law of Addition. If u, v, w e R: then (u + v) + w = u + (v + w). 41 3. Commutative Law of Addition. If u, v e R, then u + v = v + u. 41 4. Existence of Zero. There exists an element O e R such that for all v e R, v +0 = v. Cl 5. Existence of Negative. For each v e R there exists an element w (usually denoted —v) such that v + w Cl 6. Closure under Multiplication by a Scalar. If v e R and e IR, then e R. 41 7. Associative Law of Multiplication by a Scalar. If X," e IR and v e R; then A(gv) = 8. Scalar Multiplication Identity. If v e V, then Iv = v. 9. Scalar Distributive Law. If e IR and v e R, then (A + 'L)v = .Xv + 41 10. Vector Distributive Law. If e IR and u, v e R, then + v) = Au + Av. Here is a proof that R does not satisfy closure under multiplication by a scalar. 1 Consider the scalar —1 and the vector v , which belongs to R since its first component is x 1 > O. Then —1 is not in R since its first component is Thus R is not closed under multiplication by a scalar. Note that to prove a condition is not satisfied we only need to exhibit a single counter-example Be careful about how you try to construct your counter-example though, if we had chosen v or had chosen then we would not have produced a counter-example.

 

Machine generated alternative text: The previous question's discussion of counter-examples is worth amplifying. To check that a subset of S is a subspace of a vector space V, it is necessary that each of the properties of a vector space hold for S However to show that S is not a subspace we only need to find one example of a single property which does not hold. Which of these "real life" examples are analogous: [select all that apply] A car is top notch if it always starts A car is not top notch if it sometimes and sometimes doesn't Cl Saying a car is not top notch means that it never starts. To prove that a computer program has a bug, we only need to find one example where it does not work properly Cl To prove that a computer program has a bug, we need show that it never works properly To prove that your love is true, he/she ought to be faithful every day Cl To prove that your love is true, she/he only needs to be faithful once. Cl To prove that your love is untrue, she/he needs to be unfaithful every day To prove that your love is untrue, she/he needs only to be unfaithful once. Okay maybe we are pushing the analogy here, but we hope you get the idea

 

Machine generated alternative text: Let's check your understanding: Select the statements below that can be proven by finding a suitable example or counter-example, and those which require a more general proof. The question (or statment): Given a convex quadrilateral ABCD; prove that the uadrilateral formed by joining the midpoints of AB, BC, CD, and DA is a parallelogram. Is the set {O, 1, 2, 3, 4, 5, 6, 7, 8, 9} closed under dditlon? Prove your answer. Show that, in general; ATA # AAT even if A is uare_ Prove that llal — lbll la — bl for all a, b e IRn 10 here exists 10 vectors in R , each pair of which is 5 part here exist a real matrix G such that GCT _ 45 0 0 20 Suppose that v and w are vectors in IR3- and are real numbers Prove the scalar distributive law (A + = + uv and the vector distributive law can be answered (or proven) using a: general proof suitable example suitable example general proof suitable example suitable example general proof

 

Machine generated alternative text: Show that, in general; matrices cannot be cancelled from products (i.e. show that AB = AC does not necessarily imply B = C). Let Q be a square n x n orthogonal matrix. Show that e columns of Q are a set of n orthonormal vectors in Show that, in general; det(AA) # det(A). Show that, in general; a square orthogonal matrix Q has a alue for det(Q) of +1 or —1 _ suitable example ' general proof suitable example general proof

 

 

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