Transformation of a
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Calculus: Early Transcendental Functions
- Proof Prove that a the zero transformation and b the identity transformation are linear transformations.arrow_forwardProof Use the concept of a fixed point of a linear transformation T:VV. A vector u is a fixed point when T(u)=u. (a) Prove that 0 is a fixed point of a liner transformation T:VV. (b) Prove that the set of fixed points of a linear transformation T:VV is a subspace of V. (c) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(x,2y). (d) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(y,x).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
- 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 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_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_forwardGuided Proof Let B be an invertible nn matrix. Prove that the linear transformation T:Mn,nMn,n represented by T(A)=AB is an isomorphism. Getting started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8, you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is {0} Theorem 6.6. Use the fact that B is an invertible nn matrix and that T(A)=AB. (iii) Conclude that T is an isomorphism.arrow_forward
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