Fundamentals of Matrix Algebra, 2011a

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Chapter 5 Graphical Exploraons of Vectors 2. Is T(k⃗x) = kT(⃗x)? Let’s arbitrarily pick k = 7, and use ⃗x as before. ([ ]) 21 T(7⃗x) = T −14 [ ] −7 = 7 [ ] −1 = 7 1 = 7 · T(⃗x) ! So far it seems that T is indeed linear, for it worked in one example with arbitrarily chosen vectors and scalar. Now we need to try to show it is always true. Consider T(⃗x +⃗y). By the definion of T, we have T(⃗x +⃗y) = A(⃗x +⃗y). By Theorem 3, part 2 (on page 62) we state that the Distribuve Property holds for matrix mulplicaon. 13 So A(⃗x + ⃗y) = A⃗x + A⃗y. Recognize now that this last part is just T(⃗x) + T(⃗y)! We repeat the above steps, all together: T(⃗x +⃗y) = A(⃗x +⃗y) (by the definion of T in this example) = A⃗x + A⃗y (by the Distribuve Property) = T(⃗x) + T(⃗y) (again, by the definion of T) Therefore, no maer what ⃗x and ⃗y are chosen, T(⃗x + ⃗y) = T(⃗x) + T(⃗y). Thus the first part of the lineararity definion is sasfied. The second part is sasfied in a similar fashion. Let k be a scalar, and consider: T(k⃗x) = A(k⃗x) (by the definion of T is this example) = kA⃗x (by Theorem 3 part 3) = kT(⃗x) (again, by the definion of T) Since T sasfies both parts of the definion, we conclude that T is a linear transformaon. . We have seen two examples of transformaons so far, one which was not linear and one that was. One might wonder “Why is linearity important?”, which we’ll address shortly. First, consider how we proved the transformaon in Example 99 was linear. We defined T by matrix mulplicaon, that is, T(⃗x) = A⃗x. We proved T was linear using properes of matrix mulplicaon – we never considered the specific values of A! That is, we didn’t just choose a good matrix for T; any matrix A would have worked. This 13 Recall that a vector is just a special type of matrix, so this theorem applies to matrix–vector mulplicaon as well. 206
5.2 Properes of Linear Transformaons leads us to an important theorem. The first part we have essenally just proved; the second part we won’t prove, although its truth is very powerful. . Ṫheorem 21 Matrices and Linear Transformaons . 1. Define T : R n → R m by T(⃗x) = A⃗x, where A is an m × n matrix. Then T is a linear transformaon. 2. Let T : R n → R m be any linear transformaon. Then there exists an unique m×n matrix A such that T(⃗x) = A⃗x. The second part of the theorem says that all linear transformaons can be described using matrix mulplicaon. Given any linear transformaon, there is a matrix that completely defines that transformaon. This important matrix gets its own name. . Ḋefinion 30 Standard Matrix of a Linear Transformaon . Let T : R n → R m be a linear transformaon. By Theorem 21, there is a matrix A such that T(⃗x) = A⃗x. This matrix A is called the standard matrix of the linear transformaon T, and is denoted [ T ]. a a The matrix–like brackets around T suggest that the standard matrix A is a matrix “with T inside.” While exploring all of the ramificaons of Theorem 21 is outside the scope of this text, let it suffice to say that since 1) linear transformaons are very, very important in economics, science, engineering and mathemacs, and 2) the theory of matrices is well developed and easy to implement by hand and on computers, then 3) it is great news that these two concepts go hand in hand. We have already used the second part of this theorem in a small way. In the previous secon we looked at transformaons graphically and found the matrices that produced them. At the me, we didn’t realize that these transformaons were linear, but indeed they were. This brings us back to the movang example with which we started this secon. We tried to find the matrix that translated the unit square one unit to the right. Our aempt failed, and we have yet to determine why. Given our link between matrices and linear transformaons, the answer is likely “the translaon transformaon is not a linear transformaon.” While that is a true statement, it doesn’t really explain things all that well. Is there some way we could have recognized that this transformaon 207
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Fundamentals of Matrix Algebra, 2011a

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