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Stanford MATH 51 - MATH 51 Midterm 1

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MATH 51 MIDTERM I October 19, 2006Instructions:• No calculators, books, notes, or electronic devices may be used during the exam.• You have 90 minutes.• There are 7 problems, each with multiple parts. You should work quickly so as to not leaveout problems towards the end of the exam.• Write solutions on the exam sheet. If extra space is needed use the back of a page.Name:(print clearly)Signature:(for acceptance of honor code)Problem 1 (15 points)Problem 2 (15 points)Problem 3 (20 points)Problem 4 (15 points)Problem 5 (15 points)Problem 6 (10 points)Problem 7 (10 points)Total (100 points)Your TA/discussion section (circle one):Antebi (15, 18) Ayala (3, 6) Easton (14, 17)Fernanadez (2, 5) Kim (8, 11) Koytcheff (9, 12)Lo (21, 24) Rosales (26, 27) Tzeng (20, 23)Zamfir (29, 30) Schultz (51A)1. (a) Find the reduced echelon form of1 0 1 −1 00 1 2 −3 00 1 2 0 31 0 1 2 3.(b) Consider the matrix A =1 1 11 2 a1 3 bwhere a and b are real numbers. Show that thenull space of A is either {0} or a line, and give conditions on a and b that guarantee thenull space of A is a line.12. [short answer] Let A be a 3 × 4 matrix (3 rows, 4 columns) with columns a1, a2, a3, a4∈ R3and rows r1, r2, r3∈ R4, i.e. A =| | | |a1a2a3a4| | | |=− rT1−− rT2−− rT3−. Let v =1234∈ R4.(i) Express Av in terms of the columns of ajof A.(ii) Express Av in terms of the rows riof A.(iii) Can the columns of A be linearly independent?(iv) How many solutions x are there to the system Ax = 0? None, one, infinitely many, ordoes it depend on A?(v) Find all possible pairs of numbe rs (p, q) so that p is the dimension of the null space ofA and q is the dimension of the column space of A.23. A certain 3 × 4 matrix A with every entry non-zero has reduced echelon formrref(A) =1 2 0 30 0 1 −20 0 0 0.(Note that rref(A) and A are not the same matrix.)(a) Find a basis for the null space of A.(b) If the columns of A, in order, are a1, a2, a3, a4∈ R3, circle all sets of vectors belowthat give a basis for the column space of A.{a1, a2} {a1, a3} {a1, a4} {a2, a3} {a2, a4} {a3, a4}{a1, a2, a3} {a1, a2, a4} {a1, a3, a4} {a2, a3, a4}{a1, a2, a3, a4}3(3. continued)(c) Find a linear depe ndence relation b etween the columns {a1, a3, a4} of A, and explainyour answer.(d) If b = 2a1+ 3a4∈ R3, where a1and a4are the first and fourth columns of A, find allsolutions x of Ax = b. [Hint: What is one solution?]44. (a) Assume that V and W are linear subspaces of Rn. Recall that vectors a, b ∈ Rnareorthogonal if a · b = 0. Let S be the set of all vectors w ∈ W that are orthogonal toevery vector v ∈ V . Prove that S is a linear subspace of Rn.(b) Suppose L ⊂ R2is a line through 0. Let x ∈ R2be a vector not in L. Let H be the setof all vectors of the form v + tx, where v ∈ L and t ≥ 0 is a nonnegative scalar.Draw a picture of H. Is H a linear subspace of R2? If yes, prove it. If no, which of thedefining conditions for a s ubspace hold for H and which fail?55. (a) Let T : R2→ R2be a linear transformation with T21=2−3and T11=41.Find a matrix B so that Bx = T(x) for all x ∈ R2. [Hint: What is21−11?](b) Let S : R2→ R2be the transformation that rotates vectors about the origin θ radianscounterclockwise, where θ is the angle adjacent to the side of length 3 in a 3, 4, 5 righttriangle. Find a matrix A so that Ax = S(x) for all x ∈ R2.345θ(c) With T and S as in parts (a) and (b), find a matrix C so that Cx = (S ◦ T)(x) for allx ∈ R2.66. (a) Let v1, v2, v3, and v4be four vectors in R5and let T : R5→ R3be a linear transforma-tion. Explain why the four vectors {T(v1), T(v2), T(v3), T(v4)} are linearly dependent.(b) Let S : R3→ R5be another linear transformation. Explain why the four vectors{(S ◦ T)(v1), (S ◦ T)(v2), (S ◦ T)(v3), (S ◦ T)(v4)} in R5must be linearly depe ndent,where T : R5→ R3is the linear transformation from part (a).77. (a) Consider the triangle in R3with vertices A(1, 2, 3), B(2, 3, 4), and C(2, 1, 5). Find thecosine of the angle of the triangle at vertex A.(b) Suppose x, y are vectors in R10and suppose x · y = 3, kxk = 2, and kyk = 3. Find thecosine of the angle between u = x + y and v = x −


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