�MIT 3.016 Fall 2005 c� W.C Carter Lecture 6 27 Sept. 21 2005: 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 Vectors as a list of associated information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cqkcq �x = ⎛⎝ ⎞⎠ number of steps to the east number of steps to the north (6-1) number steps up vertical ladder �x = ⎛⎝ ⎞⎠ ⎛⎝ ⎞⎠ 3 2.4 xeast xnorth determines position (6-2) 1.5 xup 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 x = ⎛⎝ x y ⎞⎠ = ⎛⎝ r cos θ r sin θ ⎞⎠ (6-3) z z 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 �x = ⎛⎝ r θ ⎞⎠ = ⎛⎝ 2�x2 + ytan−1 y x ⎞⎠ (6-4) zz�MIT 3.016 Fall 2005 � W.C Carter Lecture 6c28 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 = . (6-5). . ⎝ number of Plutonium atoms . . .⎠ Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cqkcq ⎛⎜ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛⎜⎜⎜⎜⎜⎜⎜⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟number of H Navag. number of He moles of H moles of He moles of Li ⎞⎟ ⎟⎟⎟⎟⎟⎟Navag. number of Li1 Navag. �n ≡ Navag. =. = m (6-6). .. . . moles of Pu .⎝ ⎠ number of Pu Navag. . .⎝ ⎠ . . . Vector norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cqkcq ��2 2 2 x1 + x2 + . . . xk = euclidean separation (6-7)x� ≡nH+ nHe + . . . n132? = total number of atoms (6-8)��n� ≡cqkcqUnit vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .MIT 3.016 Fall 2005 � W.C Carter Lecture 6 c29 unit direction vector mole fraction composition (6-9) �x �m ˆx = ��x� ˆm = ��m� (6-10) 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, �N (e.g., each of the N entries in the x ∈ �vector of length N must be a real number—or in the set of real numbers. �� .x� ∈ �For the unit (direction) vector: ˆ ��x = {�3 x� = 1} (i.e, the unit direction vector x ∈ � |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 havex and ˆthe same number of entries, but compared to �x, the number of independent entries in ˆx 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- m ∈ (�+)(elements-1).negative numbers) and ˆMatrices and Matrix OperationsMIT 3.016 Fall 2005 � W.C Carter Lecture 6c30 Consider methane (CH4), propane (C3H8), and butane (C4H10). H-column C-column MHC = ⎛ ⎜⎜⎝ 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 ⎞ ⎟⎟⎠ methane row propane row (6-11) butane row MHC = ⎛⎝ ⎞⎠ = ⎛⎝ ⎞⎠ 4 1 8 3 10 4 M11 M12 M21 M22 (6-12) M31 M32 cqkcqMatrices as a linear transformation of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N�HC = (number of methanes, number of propanes, number of butanes) (6-13) = (NHC m, NHC p, NHC b) (6-14) = (NHC 1, NHC 2, NHC 3) (6-15) (6-16)� MIT 3.016 Fall 2005 � W.C Carter Lecture 6c31 3�N�HCM HC ij = N� (6-17)N HC iMHC ≡ i=1 The “summation” convention is often used, where a repeated index is summed over all its possible values: pN HC iM HC ij ≡ N HC iM HC ij = N j (6-18) i=1 For example, supp ose N�HC = (1. 2×1012 molecules methane, 2. 3×1013 molecules propane, 3. 4×1014 molecules butane) (6-19) N�HCM HC = × 1014 × 1013 × 1012(1. 2 methanes, 2. 3 propanes, 3. 4 butanes) ⎛ ⎜⎜⎝ 4 atoms H 1 atoms C methane methane 8 atoms H 3 atoms C propane propane 10 atoms H atoms C butane butane ⎞ ⎟⎟⎠ × 1014 × 1014=(7. 0 atoms H, 2. 0 atoms C) (6-20) cqkcqMatrix transpose operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Above the lists (or vectors) of atoms were stored as rows, often it is convenient to store them as columns. The op eration to take a row to a column (and vice-versa) is a “transp ose”. methane-column propane-column butane-column ⎛⎝ ⎞⎠ number of H number of H number of H methane molecule propane molecule butane molecule number of C number of C number of C T HC =M hydrogen row carbon row methane …