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Mass-Spring System The mass in a mass-spring system (see figure) is pulled downward and then released, causing the system to oscillate according to
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Elementary Linear Algebra (MindTap Course List)
- Calculus Let B={1,x,sinx,cosx} be a basis for a subspace W of the space of continuous functions and Dx be the differential operator on W. Find the matrix for Dx relative to the basis B. Find the range and kernel of Dx.arrow_forwardCalculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.arrow_forwardCalculus Use the matrix from Exercise 45 to evaluate Dx[4x3xex]. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.arrow_forward
- Calculus Define T:P2R by T(p)=01p(x)dx What is the kernel of T?arrow_forwardGuided proofLet u,v be the Euclidean inner product on Rn. Use the fact that u,v=uTv to prove that for any nn matrix A, (a) ATAu,v=u,Av and (b) ATAu,u=||Au||2 Getting started: To prove a and b, make use of both the properties of transposes Theorem 2.6 and the properties of the dot product Theorem 5.3. (i) To prove part a, make repeated use of the property u,v=uTv and Property 4 of Theorem 2.6. (ii) To prove part b, make use of the property u,v=uTv, Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3.arrow_forwardProof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.arrow_forward
- Guided Proof Prove that if w is orthogonal to each vector in S={v1,v2,,vn}, then w is orthogonal to every linear combination of vector in S. Getting Started: To prove that w is orthogonal to every linear combination of vectors in S, you need to show that their inner product is 0. i Write v as a linear combination of vectors, with arbitrary scalars c1,,cn in S. ii Form the inner product of w and v. iii Use the properties of inner products to rewrite the inner product w,v as a linear combination of the inner products w,vi, i=1,,n. iv Use the fact that w is orthogonal to each vector in S to lead to the conclusion that w is orthogonal to v.arrow_forwardTrue or False? In Exercises 55 and 56, 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 A set S of vectors in an inner product space V is orthonormal when every vector is a unit vector and each pair of vectors is orthogonal. b If a set of nonzero vectors S in an inner product space V is orthogonal, then S is linearly independent.arrow_forwardCalculus Repeat Exercise 45 for B={e2x,xe2x,x2e2x}. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.arrow_forward
- Guided Proof Prove that if u is orthogonal to v and w, then u is orthogonal to cv+dw for any scalars c and d. Getting Started: To prove that u is orthogonal to cv+dw, you need to show that the dot product of u and cv+dw is 0. i Rewrite the dot product of u and cv+dw as a linear combination of (uv) and (uw) using Properties 2 and 3 of Theorem 5.3. ii Use the fact that u is orthogonal to v and w, and the result of part i, to lead to the conclusion that u is orthogonal to cv+dw.arrow_forwardCAPSTONE (a) Explain how to determine whether a function defines an inner product. (b) Let u and v be vectors in an inner product space V, such that v0. Explain how to find the orthogonal projection of u onto v.arrow_forward
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