linear_algebra

linear_algebra - EE 505 B, Autumn, 2011 Linear Algebra 1 1...

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Unformatted text preview: EE 505 B, Autumn, 2011 Linear Algebra 1 1 Introduction The back cover of Gilbert Strang’s book “Introduction to Linear Algebra” summarizes all of linear algebra: A ~ x = ~ b ( N × N ) Linear systems A ~ x = ~ b ( M × N ) Least squares A ~ x = λ ~ x ( N × N ) Eigenvalues A ~ v = σ ~ u ( M × N ) Singular values If you understand these equations, then you know almost everything there is to know about linear algebra theory, and you need not continue reading. In this class, we can’t cover the whole of linear algebra (which is typically taught over a period of one or two semesters). Instead, we cover just a few concepts from linear algebra that are important to electrical engineering. 2 Matrices Definition 1. A matrix is an M × N array of numbers that has M rows and N columns. Example 2.1. A = 1 2 3 2 5 2 6- 3 1 In MATLAB , one would create this matrix using the following command: A = [1 2 3; 2 5 2; 6 -3 1] The elements of the rows are separated by spaces, and the columns delimited by semi- colons. 2.1 A Matrix is an operator A matrix (A) acts on a vector ~ x to produce the vector A ~ x . You can think of this like a system with ~ x as the input and A ~ x as the output. Example 2.2. ~ x = 1 2- 1 A ~ x = 1 2 3 2 5 2 6- 3 1 1 2- 1 EE 505 B, Autumn, 2011 Linear Algebra 2 The output will be a 3 × 1 vector. We’ll get to the mechanics of matrix multiplication in a moment. In MATLAB , one would create this vector and perform the matrix multiplications using the following commands: x = [1; 2; -1] A*x Creating the vector in MATLAB is the same as creating a 3 × 1 matrix. Example 2.3. Here is an example that illustrates the operation of a particular matrix on a vector. In this example, we derive the operation of the matrix using trigonometry. The matrix is the rotation matrix in two-dimensional space: R = cos θ- sin θ sin θ cos θ . Figure 1 shows graphically what operation the matrix performs. The matrix rotates a vector ~ v so that is becomes ~ v . x y ~ v v x v y θ 1 ~ v v x v y θ 2 θ Figure 1: The vector ~ v is rotated by θ from ~ v . To derive this matrix, we use trigonometry. v x = k ~ v k cos θ 1 v y = k ~ v k sin θ 1 and similarly, v x = k ~ v k cos θ 2 v y = k ~ v k sin θ 2 EE 505 B, Autumn, 2011 Linear Algebra 3 Now we express the equations in terms of θ . To do this, we use cos θ 2 = cos ( θ 1 + θ ) = cos θ 1 cos θ- sin θ 1 sin θ and sin θ 2 = sin ( θ 1 + θ ) = sin θ 1 cos θ + cos θ 1 sin θ . We also need and expression for k ~ v k : k ~ v k = v x cos θ 1 . Substituting these expressions into the expression for v x and v y , and using v y = v x tan θ 1 , we get v x = v x cos θ 1 ( cos θ 1 cos θ- sin θ 1 sin θ ) = v x cos θ- v y sin θ and v y = v y sin θ 1 ( sin θ 1 cos θ + cos θ 1 sin θ ) = v y cos θ + v x sin θ ....
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This note was uploaded on 03/01/2012 for the course EE 101 taught by Professor Munk during the Spring '12 term at Alaska Anch.

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linear_algebra - EE 505 B, Autumn, 2011 Linear Algebra 1 1...

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