 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 13248 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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.