 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
ISBN: 9780136009290
Solutions for Chapter 5.1: The Scalar Product in Rn
Get Full SolutionsSince 21 problems in chapter 5.1: The Scalar Product in Rn have been answered, more than 4937 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Chapter 5.1: The Scalar Product in Rn includes 21 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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!).

Fundamental Theorem.
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.

Graph G.
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.

Indefinite matrix.
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.

Multiplier eij.
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)ยท(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.

Spanning set.
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.