Kindly correctly and handwritten. 3al 34% 15:19NOTES ON CHANGE OF BASIS AND LINEAR TRANSFORMATIONSMICHAEL WOODBURYSuppose that V is a vector space of dimension n and that each of the setsS = {e1,€2, . .. , en},T = {v1, V2, ..., Vm}U = {w1, W2, ..., wn}is a basis of V.11. CHANGE OF BASIS MATRIXIn certain instances, for example if V = R" and S is the standard basis (i.e., e; is the i-thcolumn of Im), it is very easy to see how to write any element of T or U in terms of S. Thismeans that finding the coordinate vectors(1)[vi]s• [va]g•….. [v»]sand/or(2)is easy. What is harder is to find how to write T in terms of U of vice versa, or moregenerally, how to compute [x]T from [x]s or vice versa. Our goal is to come up with a matrixP, called the change of basis matrir which will do exactly that.1.1. An example. Let V = P, = {a+ bt | a,b are scalars}, and letS = {1,t}, T= {1+2, -6+8t}, u = {L+t, L-t}.w2Then the coordinate vectors of v, and vz with respect to S are[vi]s =and (v:]s =Note that[e]s =|and (e)s =||:which implies that ifA =is the matrix whose columns are [v1]s and [v1]s thenAfe) = [: ][:]=[:]=(wila-6Date: Spring 2022.andIn other words, we can think of A as being a transformation that sends the vectors T intothe vectors S. To help us remember that this is what is happening, we write(3)- A-This matrix is called the change of basis matrir from S to T.Similarly, the coordinate vectors of w1 and wz with respect to S areand (ws), - [].%3Dand so the matrix B which has [w,], and [w,], as its columns has the property that itsends the basis vectors U to S. In other words, the change of basis matrix from U to S is11 We consider the vector space R3 which has the standard basisS = {e1, e2, e3},%Dwhere e; is the i-th column of the identity matrix I3.1. Consider the set-{{} [}| }T =1(a) Explain why T is a basis of R³. Be sure to fully justify any claims/assertions thatyou make.(b) Find the matrix A which transforms T into the standard basis S. (In the textbook,this matrix is denoted_P_)SET(c) Find the matrix B which transforms the standard basis into T. (In the textbook,this matrix is denoted_P .)

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We consider the vector space R3 which has the standard basis S = {e1,€2, €3}, where e; is the i-th column of the identity matrix I3. 1. Consider the set -{{} [}] } T = 1 (a) Explain why T is a basis of R³. Be sure to fully justify any claims/assertions that you make. (b) Find the matrix A which transforms T into the standard basis S. (In the textbook, this matrix is denoted P_) SET (c) Find the matrix B which transforms the standard basis into T. (In the textbook, this matrix is denoted _P .) TES

We consider the vector space R3 which has the standard basis S = {e1,€2, €3}, where e; is the i-th column of the identity matrix I3. 1. Consider the set -{{} [}] } T = 1 (a) Explain why T is a basis of R³. Be sure to fully justify any claims/assertions that you make. (b) Find the matrix A which transforms T into the standard basis S. (In the textbook, this matrix is denoted P_) SET (c) Find the matrix B which transforms the standard basis into T. (In the textbook, this matrix is denoted _P .) TES

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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