Applied Mathematics 1411A/B Chapter 1.8.2: Applied Mathematics 1411A/B Chapter 1.8.: Applied Mathematics 1411A/B Chapter 1.8: Applied Mathematics 1411A/B Chapter 1.: Section 1.8.2

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harlequinbadger850

30 Dec 2018

School

Western University

Department

Applied Mathematics

Course

Applied Mathematics 1411A/B

Professor

Bogdan Tudose

For unlimited access to Textbook Notes, a Class+ subscription is required.

I skipped a bunch because it wasn't relavent to what I was learning in

chapter 4 and frankly it was all just proofs and theorems leading up to

this.

Theorum 1.8.4

If TA: Rn-> Rmand TB: Rn-> Rmare matrix transformations, and if

TA(x) = TB(x) for every vector x in Rn, then A = B.

This has some important consequences because it tells us that there is a

one-to-one correspondence between m xnmatrices and matrix

transformations from Rnto Rmin the sense that every mx nmatrix

produces exactly one matrix transformation (exactly one multiplication by

A) and every matrix transformation from Rnto Rmarises from exactly one

mx nmatrix.

This one matrix that comes from one transformation is said to be the

standard matrix for the transformation.

So basically, each specific transformation can only connect to one matrix

and each specific matrix can only do one transformation to convert a

vector into a new vector space.

SO. To do the specific transformation, there is one matrix that will do it.

That is the Amatrix. That is the standard matrix for the transformation.

For example, for a given linear system

That we want to transform from R2 -> R2in the sense that

The standard matrix to do this transformation for this system is

But how do we find a matrix when we are given a transformation? And

what does that look like?

The standard matrix for a given linear transformation T: Rn-> Rmis given

by the formula

Where e1, e2,... enare the standard basis vectors for Rnin column form (Rn

is the vector space which we're coming from)

Find what the standard basis vectors e1, e2,... en in Rnlook like in Rm

Build the matrix with these new basis vectors in their respective

columns

Thus it follows that the steps to finding the Standard Matrix for a Matrix

Transformation are:

Example

Find the standard matrix Afor the linear transformation T: R2 -> R2defined

by the formula

Recall that a transformation is

just a conversion of one vector

into another and this is done by

multiplying that vector by the

transformation matrix A.

This is getting the A value from the type

of transformation we're looking for

Recall the standard matrix Ais the only matrix that will perform the

desired transformation for the desired linear system

Good Definition!!!

Notes 4

April 19, 2018

6:11 PM

1.8 Matrix Transformations Page 1

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Document Summary

Recall that a transformation is just a conversion of one vector into another and this is done by multiplying that vector by the transformation matrix a. This is getting the a value from the type of transformation we"re looking for. I skipped a bunch because it wasn"t relavent to what i was learning in chapter 4 and frankly it was all just proofs and theorems leading up to this. If ta: rn -> rm and tb: rn -> rm are matrix transformations, and if. Ta (x) = tb (x) for every vector x in rn, then a = b. This one matrix that comes from one transformation is said to be the standard matrix for the transformation. So basically, each specific transformation can only connect to one matrix and each specific matrix can only do one transformation to convert a vector into a new vector space. To do the specific transformation, there is one matrix that will do it.

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This question addresses inverse transformations from a conceptual standpoint, a. Suppose that a linear transformation T: x rightarrow Ax from A to A takes some nonzero vector x to 0. Can this transformation be reversed? Does A exist? Suppose that a linear transformation T: x rightarrow Ax from is not one-to-one. What does that tell you about the set of vectors b that can be produced by applying the transformation T to all vectors x in A? How do you know, in that case, that the transformation T is not reversible (i.e. that there does not exist an inverse transformation with the standard matrix A-1 that can be applied to the result of Ax to restore the vector x to its original value)? Suppose that a linear transformation T: x rightarrow Ax from A to A is not onto. What does that tell you about the set of vectors b that can be produced by applying the transformation T to all vectors x in A? Is it possible for a linear transformation x rightarrow Ax from A to A to be one-to-one but not onto, or onto but not one-to-one? If so, please give examples of each case. If not, explain why not. Is it possible for an arbitrary linear transformation x rightarrow Ax from A to B to be one-to-one but not onto, or onto but not one-to-one? If so. please give examples of each case. If not, explain why not. For what values of {a. b. c, d. e.f. a. h. i.j } will the matrix A = invertible? what is the least amount of partitioning that needs to be applied to M; to make M1 and M2 conformable for block multiplication? What is the product M = M1, M2 that you obtain using block multiplication with these partitions? What is the product M = M1 M2. that you obtain by multiplying the two matrices without partitioning? For which three values of a is the matrix B = not invertible, and why If M1 and M2 arc partitioned matrices such that M1 For each matrix product shown below, draw in the partitions that would need to be added to make the multiplication possible: Describe in general how the sizes of each of the partitions(#rows, #columns) in a matrix product C = AB are defined by the sizes of the partitions in each of the matrices A and B when partitioned matrices are multiplied.

ey table is seen below xeecions tor Chapter 1 12, Fori, b â¬ Î.el * b-aa (0 (D 14 ForVEQ r the operation defined in Example 1 on the wr John Sue Henry. P iDFor 15ine if is ass ociative by determining if the final six engu defined ang property (ii)of Theorem 1 . So no element i if the operntion debined in Example 1 of Emle if each element has an inverse Section 1.3 Groups iw comutatityn ild Chapter 3 where the Funda the exercises that follow etermining why they are not Write out the Cayley table for the gru(o)nud lensify thee element the 10 elements of the group Z 18 its operation uses ts in the first coordinnte and +2 in the secotu Ldessity the Zn and write out the Cayley table. Reeall th of each element does Theorem 1.14 tell us aont the entries in the Cayley table ol s xrewp that is set with exactly one element. A {n), will always form group since there 19. What nly one way to define an operation on the set. Be sure to shhow that the properi combination of the usual notice what set the rule is be of a group all hold. 21. We can create even create groupe with games! Consider four cups placed in a ssare patter table. If we have a penny in one of the cups there are four ways we can move it to t cup: Horizontally, Vertically, Diagonally, or Stay nehere it is. We will p, S. To define nn operation, consider two movements in a row, v.e.,m ove the penny as r tells us to, then after that move the penny as y instruets. the set of integers the set of real numbers n the set of positive integers. e set of positive integers. n the set of positive integers. on the set of integers e set of negative integers. we ple, HV D since if we first move it horizontally and then vertically nltogether e moved it diagonally. Create the Cayley table for this group, identify the identity and the inverse of cach element 22. Prove that the set A 0 under matrix multiplication. Is it abelian? 23. Verify that 3(R) is a group using the addition of funetions as defined in Example 13 e set of nonzero rational n he set of nonzero real num n on the set R (-1 Section 1.4 Basic Algebra in Groups 24. Prove the Cancellation Law (Theorem 1.20). Do not use any theorems that occur after it in the text ssociative, commutative, has roof or counterexample fo 25. Assume G is a group and a e G. Consider a fixed integer k and use PMI (Theorem 0.3) to prove that for all nEN, a aYou will need to consider cases here since k can be positive, negative, or o 26. Prove Theorem 1.23 (ii) for the case of n > 0 and m

Applied Mathematics 1411A/B Chapter 1.8.2: Applied Mathematics 1411A/B Chapter 1.8.: Applied Mathematics 1411A/B Chapter 1.8: Applied Mathematics 1411A/B Chapter 1.: Section 1.8.2

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