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Finding
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Calculus
- Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. a Explain what it means to say that a set of vectors is linearly independent. b Determine whether the set S is linearly dependent or independent. S={(1,0,1,0),(0,3,0,1),(1,1,2,2),(3,4,1,2)}arrow_forwardMass-Spring System The mass in a mass-spring system see figure is pulled downward and then released, causing the system to oscillate according to x(t)=a1sint+a2cost where x is the displacement at time t,a1 and a2 are arbitrary constant, and is a fixed constant. Show that the set of all functions x(t) is a vector space.arrow_forwardTrue or false? In Exercises 43and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The orthogonal complement of Rn is empty set. (b) If each vector vRn can be uniquely written as a sum of a vector s1 from S1 and a vector s2 from S2, then Rn is direct sum of S1 and S2.arrow_forward
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- Question about special vectors on cubes and clocks in the figure 1.4 attached. A. Which point of the cube is i + j? Which point is the vector sum of i = (1, 0, 0) and j = (0,1,0) and k = (0,0,1)? Describe all points (x,y,z) in the cube. B. In xyz space, where is the plane of all linear combinations of i = (1,0,0) and i +j = (1,1,0)?arrow_forward4) Let two vector functions be defined by ??(?) = 〈? 2, ?, 3? 3 〉 ??(?) = 〈? − 1, ? 2 4, 5 − ?〉 a. Find the intersection point of the functions. Is it a collision point? b. Find the equation of the line that goes through the intersection point that is also orthogonal to both curves at that point.arrow_forwarda. Please provide a well explained answer for the follwoing. And explain what are the components of a vector? what if the initial point is not (0,0)? given a vector v and a scalar k, what is kv (geometrically and algebraically)? -v?arrow_forward
- Prove Theorem 1.6 (Let V be a vector space, and let S1 ⊆ S2 ⊆V. If S1 islinearly dependent, then S2 is linearly dependent.)and its corollary.arrow_forward1. Find a vector equation of the plane containing the points ?(2, 0, −1), ?(3, 4, 0), and ?(4, −1, 1).arrow_forward2 (a) Find the magnitude of the vector whose initial point is P(0, 0) and the endpoint is at Q(4, 3). (b) Write down the properties of the curl. Find the curl of any vector function at the point (1, 0, 3)arrow_forward
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