Linear Algebra

234 Chapter Three. Maps Between SpacesProof This computation is Exercise 21.QEDWe have seen here, as in the Mechanics of Matrix Multiplication subsection,that we can exploit the correspondence between linear maps and matrices. Sowe can fruitfully study both maps and matrices, translating back and forth towhichever helps the most.Over this whole section we have developed an algebra system for matrices.We can compare it with the familiar algebra system for the real numbers. Herewe are working not with numbers but with matrices. We have matrix additionand subtraction operations, and they work in much the same way as the realnumber operations, except that they only combine same-sized matrices. Wehave scalar multiplication, which is in some ways another extension of realnumber multiplication. We also have a matrix multiplication operation anda multiplicative inverse. These operations are somewhat like the familiar realnumber ones (associativity, and distributivity over addition, for example), butthere are differences (failure of commutativity). This matrix system provides anexample that algebra systems other than the elementary real number systemcan be interesting and useful.Exercises4.12 Supply the intermediate steps in Example 4.9.̌ 4.13 Use(Corollary)4.11 to(decide)if each matrix(has)an inverse.2 1 0 42 −3(a)(b)(c)−1 1 1 −3 −4 6̌ 4.14 For each invertible matrix in the prior problem, use Corollary 4.11 to find itsinverse.̌ 4.15 Find the inverse, if it exists, by using the Gauss-Jordan Method. Check theanswers for the 2×2 matrices with Corollary 4.11.⎛ ⎞( ) ( ) ( ) 1 1 33 1 2 1/22 −4(a)(b)(c)(d) ⎝ 0 2 4⎠0 2 3 1 −1 2−1 1 0⎛⎞ ⎛⎞0 1 5 2 2 3(e) ⎝0 −2 4⎠(f) ⎝1 −2 −3⎠2 3 −2 4 −2 −3̌ 4.16 What matrix has this one for its(inverse?) 1 32 54.17 How does the inverse operation interact with scalar multiplication and additionof matrices?(a) What is the inverse of rH?(b) Is (H + G) −1 = H −1 + G −1 ?̌ 4.18 Is (T k ) −1 = (T −1 ) k ?4.19 Is H −1 invertible?4.20 For each real number θ let t θ : R 2 → R 2 be represented with respect to thestandard bases by this matrix.( )cos θ − sin θsin θcos θShow that t θ1 +θ 2= t θ1 · t θ2 . Show also that t θ −1 = t −θ .