Practice Makes Perfect Linear Algebra: With 500 Exercises
William D. Clark
Language: English
Pages: 240
ISBN: 0071778438
Format: PDF / Kindle (mobi) / ePub
Expert instruction and plenty of practice to reinforce advanced math skills
- Presents concepts with application to natural sciences, engineering, economics, computer science, and other branches of mathematics
- Complementary to most linear algebra courses or as a refresher text
- More than 500 exercises and answers
- Hundreds of solved problems
- The Practice Makes Perfect series has sold more than 1 million copies worldwide
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+ y = −1 b. x − y −z =0 x+ y+z =0 2x − 4 y − z = 5 c. −3x + y + 2z = 0 2x − 4 y − z = 6 SOLUTION a. 4x − 3 y = 3 x + y = −1 Letting [A] be the augmented matrix, you have Using the reduced row-echelon form of [A] yields the unique solution x = 0, y = −1. x − y −z =0 b. x + y + z = 0 Letting [B] be the augmented matrix, you have Graphing calculators and matrices 35 Using the reduced row-echelon form of [B] yields x = 0 and y + z = 0. Thus, x = 0 and y = −z . Let y = t (an arbitrary real
by multiplying one row/column of A by a scalar k, and adding the result to another row/column, then det( B) = det( A). (Notice that this row/column transformation does not affect the determinant.) If two rows/columns of a matrix are proportional (that is, one is a scalar multiple of the other), then det( A) = 0. det(kA) = kn det( A) If A and B are matrices of the same size, then det( AB) = det( A)det( B) . 1 If det( A) ≠ 0, then det( A−1 ) = . det( A) A is invertible if and only if det( A) ≠ 0 .
there is a vector −u such that u + (− u ) = (− u ) + u = 0. vi. The scalar multiple au is in V. vii. a(u + v ) = au + av viii. (a + b )u = au + bu ix. a(bu ) = (ab )u x. 1u = u 77 Four results that follow immediately from these axioms are the following: 0u = 0 ; a0 = 0 ; − u = (−1)u; and if ku = 0, then k = 0 or u = 0. Here are examples of vector spaces. ◆ ◆ ◆ ◆ The spaces Rn for n ≥ 1 with the standard operations of vector addition and scalar multiplication are all vector spaces. The set
of projections, reflections, rotations, dilations, and contractions in the context of linear operators on R 2 and R3 and their corresponding matrix representations (Note: Recall that T : V → V is called a linear operator on V). For the linear operator T : Rn → Rn , the transformation matrix, denoted [T ], relative to the standard basis for Rn is referred to as the standard matrix for T, and T ( x ) = [T ]x , where [T ] = [T (e1 ),T (e 2 ),#,T (en )]. PROBLEM ⎛ ⎡ x ⎤⎞ ⎡ x cosθ − y sinθ ⎤ Given T
x = 2 − 12z y = 5 − 27 z , and z is a free variable 5. 6a + 6b + 3c = 5 −2b + 3c = 1 3a + 6b − 3c = −2 ⎡6 6 3 ⎢0 −2 3 ⎢ ⎢⎣ 3 6 −3 5⎤ 1 ⎥⎥ −2 R3 + R1 −2 ⎥⎦ ⎡0 −6 9 ⎢0 −2 3 ⎢ ⎢⎣ 3 6 −3 9⎤ 1 1 ⎥⎥ − R1 3 −2 ⎥⎦ ⎡0 2 −3 ⎢0 −2 3 ⎢ ⎢⎣ 3 6 −3 −3 ⎤ 1 ⎥⎥ 1R2 + R1 −2 ⎥⎦ ⎡0 0 0 ⎢0 −2 3 ⎢ ⎢⎣ 3 6 −3 No solution. 2x + 2 y + 4z = 2 − 3z = −9 6. −3x 6 x + 3 y + 9z = 15 ⎡2 2 4 ⎢ −3 0 −3 ⎢ ⎢⎣ 6 3 9 1 R1 2⎤2 1 −9 ⎥⎥ − R2 3 15 ⎥⎦ 1 R3 3 No solution. ⎡1 0 0 7. ⎢⎢0 1 0 ⎢⎣0 0 1 −3 ⎤ 0 ⎥⎥ 7 ⎥⎦ x = −3, y = 0,
