for part 5 chose one a or b

1. The following matrices are related. A' is row-equivalent to A, and A" is the transpose of A. Use this information to complete the following: N 2 1 -6 00 Co its co 3 A = AT = 7 A' 10 10 O 3 4 0 O O 8 -10 (a) Determine dim(A). Show or explain how you found your result. (b) Determine col(A). Show or explain how you found your result. (c) Determine row(A). Show or explain how you found your result. (d) Determine null(A). Show or explain how you found your result. (e) Determine rank(A). Show or explain how you found your result. (f) Determine nullity(A). Show or explain how you found your result. 2. Let M2x2 be the vector space of all 2 x 2 matrices with real number components over the field R. Let V be the subset of M2x2, V Since both matrix and vector addition and scalar multiplication are similarly defined (i.e. the are performed component-wise), we can think of these matrices as vectors comprised of the entries in the matrix as long as we keep the order consistent. e.g. V' (a) Is V linearly independent? Show or explain how you arrived at this conclusion. (b) Does V span M2x2? Show or explain how you arrived at this conclusion. (c) Is V a basis for M2x2? If so, show or explain how you arrived at this conclusion; if not, explain why not and find a basis (other than the standard basis) for R spanned by V. Show or explain how you arrived at this answer. (d) State the dimension of the space spanned by the vectors you described in part (c). Show or explain how you arrived at this answer.3. Is 2+ (the set of positive integers) a eld? Explain how you determined your answer. 4. Consider the unit circle, measured in radians. Let the set U be the set of points along the unit circle such that a vector u E U looks like u = ((1,5) where a, b E R. Recall that another way to write the point (c,b) is (cos a9,sin 6). Let two vectors in the set be u = (a, b) = (cos 6, sin 9) and v = (c, d) = (cos,sin ca) Dene addition in this set as u + v = 6 + .* Does U over 1R form a vector space? If so, show the way in which each criterion of a vector space is met. If not, describe which criteria of vector spaces are met and which are not. You are not required to write a proof, but you should still use general elements of the set to make your case rather than specic elements. *FOOTNOTE: This can easily be derived from the formula for arclength. Since 31 = r6 for the length of the are from (0,0) to the point ((1,5) and 32 = rqb for the length of the are from (U, U) to the point (e, d), and since, being on the unit circle, the radius is r = 1, this leaves .91 = 9 and .92 = gt. We are saying that \"addition\" in this setting is defined as the addition of distances from (0,0) around the unit circle. 5. Select one of the following two proofs to complete. If you do both, I will randomly select one to review, so you are better off writing up the one you feel you understand better. Although both proofs have been largely set up for you, my expectation here when re- viewing is to read a complete proof, not just your answers to the \"ll-in-the-blanks\". Do not submit a paper that just says \"Yes, yes, no\" for example. Write out the entire proof. Start your work from the line that starts with the part letter of the proof you are submitting; you do not need to copy out the \"Restated...\" paragraph, or the line \"I'll get you started\". a. (a) Let W be a subset of R3 such that u E W = b , with a,b,c E R. c Restated, this means: A subset of R3 called W contains vectors all of which look like u, and the components of u are real numbers. Either prove or disprove that W is a subspace of R3. I'll get you started. Proof: Recall that a subset W of vectors of a vector space V form a subspace if three criteria are met: 1. The zero vector 0 = [0,0,. . . ,0] is in W. 2. W is closed under vector addition 3. W is closed under scalar multiplication. To prove that W is not a subspace of R3, we need only show that one of the criteria does not hold. As we check each one in turn until we nd one that does not hold, if we nd after all that they all do hold, then we have proved that W is a subspace of R3. (Choose the expression in [brackets] that makes each statement true, and explain your choice.) Check: The zero vector 0 = [0,0, . . . ,0] [is]/[is not] in W. (For a proof, you must describe this vector, not just state that it is in the set; or describe why it is not in the set.) Check: W [is] / [is not] closed under vector addition. (For a proof, you must demonstrate this, one way or the other, not just state it.) Check: W [is] f [is not] closed under scalar multiplication. (For a proof, you must demonstrate this, one way or the other, not just state it.) Since [all] f [not all] of the necessary criteria are met, I have shown that W [is] / [is not] a subspace of R3. end of proof (b) ProvethatVu, vElR\" andVa ER, a(u+v)=au+av. Restated, this means: Prove that any two vectors with real number components, regardless of their size or dimension (as long as they are appropriately matched), together with any real number scalar a satisfy the equality a(u + v) = an + av. I'll get you started. Proof: Let u, v be vectors in IR\" such that u = [u1,u2,. . . , an], and v = [01,Ug,...,vn]. Then on = [ ] and av = [ ], which means that au+av=[ ] || lI Il (continue as many lines as you. need.)