- 5.1.1: Find the angle between the vectors v and w in each of the following...
- 5.1.2: For each pair of vectors in Exercise 1, find the scalar projection ...
- 5.1.3: For each of the following pairs of vectors x and y, find the vector...
- 5.1.4: Let x and y be linearly independent vectors in R2. If _x_ = 2 and _...
- 5.1.5: Find the point on the line y = 2x that is closest to the point (5, 2).
- 5.1.6: Find the point on the line y = 2x +1 that is closest to the point (...
- 5.1.7: Find the distance from the point (1, 2) to the line 4x 3y = 0.
- 5.1.8: In each of the following, find the equation of the plane normal to ...
- 5.1.9: Find the equation of the plane that passes through the points P1 = ...
- 5.1.10: Find the distance from the point (1, 1, 1) to the plane 2x + 2y + z...
- 5.1.11: Find the distance from the point (2, 1,2) to the plane 6(x 1) + 2(y...
- 5.1.12: Prove that if x = (x1, x2)T , y = (y1, y2)T, and z = (z1, z2)T are ...
- 5.1.13: Show that if u and v are any vectors in R2, then _u + v_2 (_u_ + _v...
- 5.1.14: Let x1, x2, and x3 be vectors in R3. If x1 x2 and x2 x3, is it nece...
- 5.1.15: Let A be a 2 2 matrix with linearly independent column vectors a1 a...
- 5.1.16: If x and y are linearly independent vectors in R3, then they can be...
- 5.1.17: Let x = 4 4 4 4 and y = 4 2 2 1 (a) Determine the angle between x a...
- 5.1.18: Let x and y be vectors in Rn and define p = xT y yT y y and z = x p...
- 5.1.19: Use the database matrix U from Application 1 and search for the key...
- 5.1.20: Five students in an elementary school take aptitude tests in Englis...
- 5.1.21: Let t be a fixed real number and let c = cos t, s = sin t, x = (c, ...
Solutions for Chapter 5.1: The Scalar Product in Rn
Full solutions for Linear Algebra with Applications | 8th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.