24. a. Not every orthogonal set in R is linearly independent b. If a set S he property that u whenever i j. then S is an orthonormal set. c. If the columns of an m x n matrix A are orthonormal, then the linear mapping x H Ax preserves.
(3) Answer each of the following:
(a) Disprove: If n2 is a multiple of 9, then n is a multiple of 9. [5 marks] (b) Prove: If n2 is a multiple of 3, then n is a multiple of 3. [5 marks]
(Hint: RecallthatifaZisnotamultipleof3,thena=3b+1ora=3b+2for.
3. In your own words, explain what the nullspace and column space of a matrix are, How are these related to the kernel and range oialinear transformation Uwe definitions from section 4.2 of the textbook as needed to aid in your explanation..
3. Find a vector that is orthogonal to both u = (2, 3, 1) and v = (0, 5, 7) using cross
product.
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4. If u = (1, 2, 2), v = (3, 0, 4) and w = (6, 3,.
(22) The linear transformation T y T is one-to-one
(23) The linear transformation in question 22 is onto.
(24) If T is the linear transformation in question 22, then T12 -T 1.
(25) 1 is an eigenvalue of the linear transformation.
(3 times 9 marks) Mark each statement TRUE or FALSE. [] Every matrix is row equivalent to a unique matrix in echelon form [] lf A is an m times n matrix and the equation Ax-b is consistent for every b in R", then A has m pivot columns. [].
2. Suppose that a and b are nonzero vectors, a. Under what circumstances is comp a b = comp b - a? b. Under what circumstances is proj a b - proj b a?
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(3) Use the Generalized PMI to prove the following. Also, show that for some n EN, these statements are false. a) n3 < n! for all n6 (b) (n +1)! > 2n+3 for all n 2 5.
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2. True or false? Justify. (a) Two n x n matrices A and B are similar if there exists a linear transformation T R. RT and bases A and B such that TTA TIB. (b) Let (1), (0 1 0 Let S- and i aiei a2e2. Then S v B.
2.ou have $600 in your checking account. You spend 1/3 of the $600 on a new muffler and 2/5 of the $600 on engine repairs. How much money isLEFT in your account? Express your answer in whole dollars.
please answer all questions please.
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3. For each of the following space curves, (i) find parametric equations that represent the curve, using t as the parameter and (ii) find (ds/dt) where s stands for arc length. (a) the intersection curve of z=(4x^2+y^2)^(1/2) and y=3 (b) the intersection curve of z=(4x^2+y^2)^(1/2) and z=2..
28.3
Use Fermat's little theorem to show that for p a prime, every integer a with a not identical to 0 (mod p) has a reciprocal modulo p. In Problem 28.3 above you proved that if p is a prime, then every
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2. You have been asked to come up with a design of a 2-dof planar linkage with revolute joints and binary links. What is the minimum number of links that you will need? Show all your work in deriving the number.
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3. Deriving the inconsistency in OLS Which of the following most accuralynes the inconsistancy in OLS? Covtxu im = Covxu O plim = Covxu Positive Negative or Negligible? Suppose that x, and u are positively correlated. Then the inconsistency in is Consider the following true model, which satisfies the first.
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