- Chapter 1: Starting with MATLAB
- Chapter 10: Three-Dimensional Plots
- Chapter 11: Symbolic Math All
- Chapter 2: Creating Arrays
- Chapter 3: Mathematical Operations with Arrays
- Chapter 4: Using Script Files and Managing Data
- Chapter 5: Two-Dimensional Plots
- Chapter 6: Programming in MATLAB
- Chapter 7: User-Defined Functions and Function Files
- Chapter 8: Polynomials, Curve Fitting, and Interpolation
- Chapter 9: Applications in Numerical Analysis Numerical
MATLAB: An Introduction with Applications 5th Edition - Solutions by Chapter
Full solutions for MATLAB: An Introduction with Applications | 5th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Invert A by row operations on [A I] to reach [I A-I].
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).