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Chapter 3: Linear Algebra I. Solving sets of linear equations ex: solve for x, y, z.

Chapter 3: Linear Algebra I. Solving sets of linear equations ex: solve for x, y, z. 3x + 5y + 2z = -4 2x + 9z = 12 4y + 2z = 3 (can solve longhand) (can solve same problem using matrix algebra tricks) ex: Boas- see transparency. More commonly:.

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Chapter 3: Linear Algebra I. Solving sets of linear equations ex: solve for x, y, z.

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  1. Chapter 3: Linear Algebra I. Solving sets of linear equations ex: solve for x, y, z. 3x + 5y + 2z = -4 2x + 9z = 12 4y + 2z = 3 (can solve longhand) (can solve same problem using matrix algebra tricks) ex: Boas- see transparency. More commonly:

  2. Allowed Moves: “Row operations” • Exchanging two rows (not columns!) • Multiply or divide a row by a nonzero constant. • Add or subtract one row from another. • ex: Pre-class assignment • ex: Circuit- see transparency and pg. 2

  3. ex: Circuit: Find i1, i2, i3. (Halliday and Resnick, Ch. 28, 33P)

  4. ex: Circuit (continued)

  5. II. Determinants • (only works for square matrices) • Notation: • We can extract much useful information from a matrix by boiling it down to • one number called a determinant. • A. To find the determinant: • 1) 2x2 matrix:

  6. 2) 3x3 matrix: • You can do this with any row of column. • ex: • *each person gets a different row or column* • Find det(M)

  7. 3) 4x4 matrix: analogy to 3x3. And so on… Useful facts: transparency Do examples illustrating each – base on previous example. Can use these to simplify finding determinants. ex: same Preclass Q1

  8. B. Cramer’s Rule Say we have a system of equations: (e.g. 2 equations and 2 unknowns.) The solutions for x and y are: Where (this generalizes any n equations with n unknowns.)

  9. ex: find? Preclass Q2

  10. III. Matrix Operations Let 1) Dimension:(# rows) x (# columns) dim (M1) = 3x2 , dim(M3) = 2x3 2) Equality: Note: a) Matrices must be same size (same dimension). b) This is really a set of mxn equations (aij=bij). c) Row reduction does not give equal matrices.

  11. 3) Transpose: (Exchange rows and columns.) Then 4) Multiplying by a scalar: ex: 5) Adding matrices: ex: Note: can’t add M1 and M3 because they aren’t the same dimension.

  12. 6) Multiplying Matrices: (nxm matrix) x (mxn matrix) = (nxn matrix) ex: ijth element [M1M3]ij = Multiply row i by column j and add up terms.

  13. 7) Special Matrices: • Unit matrix: All diagonal terms are 1, and all others are 0. • (square nxn matrix) • Note: I·M=M·I=M for any matrix M of the same dimension as I. • Diagonal matrix: • Upper diagonal: • Lower diagonal:

  14. 8) Inverse of a matrix: M-1 of a square matrix M is defined by M-1·M=1 and M·M-1=1. Not all matrices M have an inverse M-1. Finding M-1 is a trick! Mathematica or (tediously) by hand. By hand: M is square, so we can find det(M). Then where C is the matrix of cofactors Cij of elements Mij. defn: Cofactor Cij of Mij is (-1)I+j • ex: determinant of matrix remaining when row I and column j are crossed out of M.

  15. ex: Find M-1

  16. IV. Examples 1) 3x + 2y + z = -3 x + 2z = 1 2x + y = 4 We write Ax = b To solve for x: Ax = b A-1Ax = A-1b x = A-1 b

  17. ex: eliz’s project laser  Two positions: Measure Tsurf, Ths, Tamb at each. Can write down equation for each slice relating 3 temps.  20 coupled equations! Write in the form: AP = Ts  P = A-1Ts (Matlab solves in 30 seconds.)

  18. (-x,y) (x,y) 2) Geometry: Reflection (reflects about x-axis) (reflects about the y-axis) y x ?

  19. 3) Geometry: Rotation of coordinates (rotates coordinates by θ) ex: Say I reflect (3,2) about the x-axis and then the y-axis. Then what are it’s coordinates if I use a new coordinate system rotate by /6? y (x’,y’) y’ (x,y) x’ θ x

  20. θ2 θ1 y2 y1 d Lens (focal lenth f1) Lens (focal lenth f2) 4) Geometric Optics Describe each ray by height y and angle θ. Given (y1, θ1), what is (y2, θ2) at the output?

  21. ex: Propagating through free space θ2 θ1 d

  22. θ1 θ2 ex: Refraction at boundary y2 y1

  23. (-x,y) (x,y) V. Eigenvectors & Eigenvalues For a given operator (matrix) M, are there any vectors that are left unchanged (except for scaling the length) by M? eg: where λ is a constant ex: Reflection at about the y-axis Eigenvectors [K1,0] , [0,K2] where Eigenvalues λ=-1 λ=1

  24. y (x,y) θ x ex: Rotation If θ = 180o, Eigenvectors: all [x,y], eigenvalue λ=-1 If θ = 360o, Eigenvectors: all [x,y], eigenvalue λ=1 If θ is any other value, there are no eigenvectors & eigenvalues

  25. More formally: To find eigenvalues: Characteristic equation: Solve for eigenvalues ; then you can get the eigenvectors

  26. So, applying this to our examples: ex: Reflection about y-axis To find the eigenvalues: Characteristic equation: What are the eigenvectors?

  27. ex: Rotation Eigenvalues: Eigenvectors:

  28. ex: Find eigenvalues and eigenvectors

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