Read e-book online Linear algebra c-1 - Linear equations, matrices and PDF

By Mejlbro L.

Show description

Read or Download Linear algebra c-1 - Linear equations, matrices and determinants PDF

Best study guides books

Download e-book for iPad: Outline of Genetics by William D. Stansfield

In Genetics, 3rd version, scholars study the underlying recommendations and functions of genetics. they'll additionally locate updated assurance of molecular genetics and the molecular biology of eucaryotic cells and their viruses. a number of illustrations, 209 difficulties solved step by step, 1,000 extra perform difficulties, and 433 evaluate questions make key issues memorable and get ready readers for all types of usual genetics assessments.

Get Cay, the Reading Guide PDF

Suffering readers often lack simple studying talents and aren't built with earlier wisdom and interpreting ideas to completely interact within the school room literature event. provide your scholars the historical past and help they should comprehend and luxuriate in literature. With those examining publications, your scholars will perform analyzing comprehension talents, sharpen their vocabulary, and discover ways to determine literary components.

Get Cliffsnotes French Lieutenant's Woman (Cliffs Notes) PDF

Fowles tells an excellent love tale in regards to the results of Victorian English society on an individual's self-awareness and the way that understanding dominates and distorts one's existence, together with relationships with folks.

Additional info for Linear algebra c-1 - Linear equations, matrices and determinants

Sample text

2. (a) We shall solve the equation ⎛ ⎞ x1 1 2 4 3 ⎜ x2 AX = ⎝ 2 4 3 6 ⎠ ⎜ ⎝ x3 3 6 2 9 x4 ⎛ The total matrix ⎛ 1 2 4 ⎝ 2 4 3 3 6 2 ⎞ ⎛ ⎞ 7 ⎟ ⎟ = b1 = ⎝ 4 ⎠ . ⎠ 1 is equivalent to ⎞ ⎛ 7 ∼ 3 1 2 4 3 4 ⎠ R2 := 2R1 − R2 ⎝ 0 0 5 0 6 1 9 R3 := 3R1 − R3 0 0 10 0 ⎞ 7 10 ⎠ , 20 corresponding to the system x1 + 2x2 + 4x3 x3 + 3x4 = 7, = 2, thus x3 = 2 and x1 + 2x2 + 3x4 = −2. Choosing x2 = s and x4 = t, the set of solutions is {(−1 − 2s − 3t, s, 2, t) | s, t ∈ R}. (b) In this case ⎛ 1 ⎝ 2 3 the total matrix is equivalent to ⎞ ⎛ 7 ∼ 2 4 3 1 2 4 3 4 ⎠ R2 := 2R1 − R2 ⎝ 0 0 5 0 4 3 6 2 R3 := 3R1 − R3 6 2 9 0 0 10 0 ⎞ 7 10 ⎠ , 19 and the set of solutions is the empty set.

11 ⎛ 1 A=⎝ 4 7 2. Matrices Solve the matrix equation AX = 0, where ⎞ 2 3 5 6 ⎠ 8 9 and 0 is the (3 × 3) zero matrix. Find the rank of any solution matrix X. Since the zero matrix of coefficients ⎛ ⎞ 1 2 3 ⎝ 4 5 6 ⎠ ⎛ 7 8 9 ⎞ 1 2 3 ⎝ 3 3 3 ⎠ ⎛ 3 3 3 ⎞ 1 2 3 ⎝ 1 1 1 ⎠ ⎛ 0 0 0 ⎞ 1 2 3 ⎝ 0 1 2 ⎠ 0 0 0 consists of three zero columns, we consider the equivalent matrix of the matrix ∼ R2 R3 ∼ R2 R3 := R2 − R1 := R3 − R2 := R2 /3 := R2 − R3 ∼ R := R1 − R2 ∼ R1 := R1 − R2 ⎛ ⎞ 1 1 1 ⎝ 0 1 2 ⎠. 0 0 0 Hence a particular column in the X matrix must necessarily satisfy x1 + x2 x2 + x3 + 2x3 = 0, = 0.

0 1 It follows that the unique solution is ⎛ ⎞ −1 −3 1 ⎠. com 55 Linear Algebra Examples c-1 2. 25 Given the matrices ⎛ ⎞ ⎛ ⎞ 1 2 −2 2 6 4 a ⎠, A=⎝ 1 B = ⎝ b 13 + a ⎠ , 0 −2 1 −2 −4 where a and b are arbitrary real numbers. 1. Find the pairs (a, b), for which the matrix equation AX = B has precisely one solution. 2. Find the complete solution of the matrix equation for (a, b) = (−3, 4). 1. The total matrix is equivalent to ⎞ ⎛ 2 6 1 2 −2 ∼ b 13 + a ⎠ 4 a (A | B) = ⎝ 1 R2 := R2 − R1 −2 ⎞−4 0 −2 1 ⎛ 2 6 1 2 −2 ∼ ⎝ 0 b−2 7+a ⎠ 2 a+2 R3 := R2 + R3 −2 −4 ⎞ 1 ⎛ 0 −2 2 6 1 2 −2 ⎝ 0 2 a+2 b−2 7+a ⎠ b−4 3+a 0 0 a+3 It follows that the matrix of coefficients is of rank 3 for a = −3, hence there exists precisely one solution, when a = −3 and b ∈ R.

Download PDF sample

Linear algebra c-1 - Linear equations, matrices and determinants by Mejlbro L.


by Charles
4.5

Rated 4.72 of 5 – based on 42 votes