 3.2.1: In Exercises 12, find the norm of v, a unit vector that has the sam...
 3.2.2: In Exercises 12, find the norm of v, a unit vector that has the sam...
 3.2.3: In Exercises 34, evaluate the given expression with and .
 3.2.4: In Exercises 34, evaluate the given expression with and .
 3.2.5: In Exercises 56, evaluate the given expression with and
 3.2.6: In Exercises 56, evaluate the given expression with and
 3.2.7: Let . Find all scalars k such that .
 3.2.8: Let . Find all scalars k such that .
 3.2.9: In Exercises 910, find and .
 3.2.10: In Exercises 910, find and .
 3.2.11: In Exercises 1112, find the Euclidean distance between u and v.
 3.2.12: In Exercises 1112, find the Euclidean distance between u and v.
 3.2.13: Find the cosine of the angle between the vectors in each part of Ex...
 3.2.14: Find the cosine of the angle between the vectors in each part of Ex...
 3.2.15: Suppose that a vector a in the xyplane has a length of 9 units and...
 3.2.16: Suppose that a vector a in the xyplane points in a direction that ...
 3.2.17: In Exercises 1718, determine whether the expression makes sense mat...
 3.2.18: In Exercises 1718, determine whether the expression makes sense mat...
 3.2.19: Find a unit vector that has the same direction as the given vector....
 3.2.20: Find a unit vector that is oppositely directed to the given vector....
 3.2.21: State a procedure for finding a vector of a specified length m that...
 3.2.22: . If and , what are the largest and smallest values possible for ? ...
 3.2.23: Find the cosine of the angle between u and v. (a) (b) (c) (d)
 3.2.24: Find the radian measure of the angle (with ) between u and v. (a) a...
 3.2.25: In Exercises 2526, verify that the CauchySchwarz inequality holds.
 3.2.26: In Exercises 2526, verify that the CauchySchwarz inequality holds.
 3.2.27: Let and . Describe the set of all points for which .
 3.2.28: (a) Show that the components of the vector in Figure Ex28a are and...
 3.2.29: Prove parts (a) and (b) of Theorem 3.2.1.
 3.2.30: Prove parts (a) and (c) of Theorem 3.2.3.
 3.2.31: Prove parts (d) and (e) of Theorem 3.2.3.
 3.2.32: Under what conditions will the triangle inequality (Theorem 3.2.5a)...
 3.2.33: What can you say about two nonzero vectors, u and v, that satisfy t...
 3.2.34: (a) What relationship must hold for the point to be equidistant fro...
 3.2.a: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.b: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.c: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.d: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.e: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.f: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.g: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.h: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.i: In parts (a)(j) determine whether the statement is true or false, a...
 3.2.j: In parts (a)(j) determine whether the statement is true or false, a...
Solutions for Chapter 3.2: Norm, Dot Product, and Distance in Rn
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 3.2: Norm, Dot Product, and Distance in Rn
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. Chapter 3.2: Norm, Dot Product, and Distance in Rn includes 44 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 44 problems in chapter 3.2: Norm, Dot Product, and Distance in Rn have been answered, more than 14358 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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 .

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.