3.016 Mathematics for Materials Scientists and Engineers, Lecture 6 Notes

Lecture 6: Linear Algebra I Reading: Kreyszig Sections: §6.1 (pp:304–09) , §6.2 (pp:312–18) , §6.3 (pp:321–23) , §6.4 (pp:331–36) Vectors cq cqk Vectors as a list of associated information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . determines position (6­2) The vector above is just one example of a position vector. We could also use coordinate systems that differ from the Cartesian (x, y, z) to represent the location. For example, the location in cylindrical coordinate system could be written as as a Cartesian vector in terms of the cylindrical coordinates (r, θ, z). The position could also be written as a cylindrical, or polar vector (6­4) where the last term is the polar vector in terms of the Cartesian coordinates. Similar rules would apply for other coordinate systems like spherical, elliptic, etc. However, vectors need not represent position at all, for example: ⎛ ⎞ number of Hydrogen atoms ⎜ number of Helium atoms ⎟ ⎜ ⎟ ⎜ number of Lithum atoms ⎟ ⎜ . ⎟ �n = ⎜ . (6­5) ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ number of Plutonium atoms ⎠ ... cq cqk Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎛ number of H Navag. ⎜ ⎜ number of He ⎜ Navag. ⎜ ⎜ number of Li ⎜ 1 Navag. �n ≡ ⎜ ⎜ . Navag. ⎜ .. ⎜ ⎜ number of Pu ⎜ Navag. ⎝ .. . ⎞ ⎟ ⎛ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎟ ⎟ ⎠ ⎞ moles of H moles of He ⎟ ⎟ moles of Li ⎟ ⎟ . � ⎟=m .. ⎟ ⎟ moles of Pu ⎠ .. . (6­6) cqk cq Vector norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ��x� ≡x21 + x22 + . . . xk2 = euclidean separation ��n� ≡nH + nHe + . . . n132? = total number of atoms (6­7) (6­8) cqk cq Unit vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unit direction vector �x x̂ = ��x� mole fraction composition m � m̂ = �m� � (6­9) (6­10) Extra Information and Notes Potentially interesting but currently unnecessary If � stands for the set of all real numbers (i.e., 0, −1.6, π/2, etc.), then can use a shorthand to specify the position vector, �x ∈ �N (e.g., each of the N entries in the vector of length N must be a real number—or in the set of real numbers. ��x� ∈ �. For the unit (direction) vector: xˆ = {�x ∈ �3 | ��x� = 1} (i.e, the unit direction vector is the set of all position vectors such that their length is unity—­or, the unit direction vector is the subset of all position vectors that lie on the unit sphere. �x and xˆ have the same number of entries, but compared to �x, the number of independent entries in x̂ is smaller by one. For the case of the composition vector, it is strange to consider the case of a negative number of atoms, so the mole fraction vector �n ∈ (�+ )elements (�+ is the real non­ negative numbers) and m ˆ ∈ (�+ )(elements­1) . Matrices and Matrix Operations Consider methane (CH4 ), propane (C3 H8 ), and butane (C4 H10 ). H­column C­column ⎞ number of H number of C methane molecule methane molecule ⎟ = ⎜ number of H number of C ⎜ ⎟ ⎝ propane molecule propane molecule ⎠ number of H number of C butane molecule butane molecule ⎛ ⎞ ⎛ M11 M12 4 1 MHC = ⎝ 8 3 ⎠ = ⎝ M21 M22 M31 M32 10 4 ⎛ MHC methane row propane row butane row (6­11) ⎞ ⎠ (6­12) cqk cq Matrices as a linear transformation of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N�HC = (number of methanes, number of propanes, number of butanes) = (NHC m , NHC p , NHC b ) = (NHC 1 , NHC 2 , NHC 3 ) (6­13) (6­14) (6­15) (6­16) N�HC MHC ≡ 3 � � NHC i MHC ij = N (6­17) i=1 The “summation” convention is often used, where a repeated index is summed over all its possible values: p � NHC i MHC ij ≡ NHC i MHC ij = Nj (6­18) i=1 For example, suppose N�HC = (1.2×1012 molecules methane, 2.3×1013 molecules propane, 3.4×1014 molecules butane) (6­19) N�HC MHC = (1.2 × 1014 methanes, 2.3 × 1013 propanes, 3.4 × 1012 ⎛ 4 atoms H ⎜ 8methane atoms H butanes) ⎜ ⎝ propane 10 atoms H butane ⎞ 1 atoms C methane 3 atoms C ⎟ ⎟ propane ⎠ atoms C butane =(7.0 × 1014 atoms H, 2.0 × 1014 atoms C) (6­20) cqk cq Matrix transpose operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Above the lists (or vectors) of atoms were stored as rows, often it is convenient to store them as columns. The operation to take a row to a column (and vice­versa) is a “transpose”. methane­column propane­column butane­column ⎞ number of H number of H number of H = ⎝ methane molecule propane molecule butane molecule ⎠ hydrogen row number of C number of C number of C carbon row methane molecule propane molecule butane molecule (6­21) ⎛ ⎞ ⎞ ⎛ number of methanes NHC m T N�HC = ⎝ number of propanes ⎠ = ⎝ NHC p ⎠ (6­22) number of butanes NHC b ⎛ MHC T T �T MHC T N�HC = N � 4 8 10 1 3 4 � ⎛ ⎞ � � number of methanes ⎝ number of propanes ⎠ = number of H­atoms number of C­atoms number of butanes (6­23) cq cqk Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r Mathematica� Example: Lecture­06 Matrices Suppose that some process that produces hyrdocarbons can be modeled with the pressure P and temperature T . Suppose (this is an artificial example) that the number of hydrocarbons produced in one millisecond can be related linearly to the pressure and temperature: Creating a Matrix number of methanes = αP + βT number of propanes = γP + δT number of butanes = �P + φT (6­24) or ⎛ T N�HC ⎞ � � α β P =⎝ γ δ ⎠ T � φ (6­25) Then, if we wanted to find an operation that takes us from the processing vector (P, T )T to the number of hydrogens and carbons: Matrix multiplication � Q P T ⎛ � = MHC T ⎞ � � � � α β P number of H­atoms ⎝ γ δ ⎠ = T number of C­atoms � φ Using matrix multiplication, � � 4α + 8γ + 10� 4β + 8δ + 10φ Q= α + 3γ + 4� β + 3δ + 4φ (6­26) (6­27) is a matrix, which when operating on a vector of pressure and temperature, returns a vector of the amount of hydrogen and carbon. Matrix multiplication is defined by: AB = � Aki Bij (6­28) i The indices of the matrix defined by the multiplication AB = C are Ckj . cqk cq Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sometimes what we wish to know, “What vector is it (�x), when transformed by some matrix (A) gives us a particular result (�b = A�x)?” A�x = �b A−1 A�x = A−1�b �x = A−1�b (6­29) The inverse of a matrix is defined as something that when multiplied with the matrix leaves a product that has no effect on any vector. This special product matrix is called the identidy matrix. r Mathematica� Example: Lecture­04 Inverting Matrices Using Inverse[] Q −1 1 = det(Q) � −(β + 3δ + 4φ) −(2β + 4δ + 5φ −(α + 3γ + 4�) −(2α + 4γ + 5� � (6­30) where det(Q) ≡= 4(αδ − βγ) − 6(β� − αφ) + 2(γφ − δ�) (6­31) cqk cq Linear Independence: When solutions exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � a11 a12 a21 a22 �� A�x = �b � � � x b1 = b2 y (6­32) b1 b2 a12 a22 a11 x + a12 y = b1 a11 x + a12 y = b1 a11 x + a12 y = b1 a21 x + a22 y = b2 a21 x + a22 y = b2 No Solution One Unique Solution a21 x + a22 y = b2 Infinitely Many Solutions Figure 6­1: Geometric interpretation of solutions in two dimensions r Mathematica� Example: Lecture­06 Eliminating redundant equations or variables Consider liquid water near the freezing point—dipole interactions will tend to make water molecules form clusters such as H2 O and H4 O2 . Then the mapping from molecules to the number of atoms becomes: � �� � � � NH2 O 2 4 NH = (6­33) 1 2 NH4 O2 NO Using RowReduce[] cq Linear dependence and the rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cqk