 4.1.1: Let V be the set of all ordered pairs of real numbers, and consider...
 4.1.2: Let V be the set of all ordered pairs of real numbers, and consider...
 4.1.3: In Exercises 312, determine whether each set equipped with the give...
 4.1.4: In Exercises 312, determine whether each set equipped with the give...
 4.1.5: In Exercises 312, determine whether each set equipped with the give...
 4.1.6: In Exercises 312, determine whether each set equipped with the give...
 4.1.7: In Exercises 312, determine whether each set equipped with the give...
 4.1.8: In Exercises 312, determine whether each set equipped with the give...
 4.1.9: In Exercises 312, determine whether each set equipped with the give...
 4.1.10: In Exercises 312, determine whether each set equipped with the give...
 4.1.11: In Exercises 312, determine whether each set equipped with the give...
 4.1.12: In Exercises 312, determine whether each set equipped with the give...
 4.1.13: Verify Axioms 3, 7, 8, and 9 for the vector space given in Example 4.
 4.1.14: Verify Axioms 1, 2, 3, 7, 8, 9, and 10 for the vector space given i...
 4.1.15: With the addition and scalar multiplication operations defined in E...
 4.1.16: Verify Axioms 1, 2, 3, 6, 8, 9, and 10 for the vector space given i...
 4.1.17: Show that the set of all points in lying on a line is a vector spac...
 4.1.18: Show that the set of all points in lying in a plane is a vector spa...
 4.1.19: In Exercises 1921, prove that the given set with the stated operati...
 4.1.20: In Exercises 1921, prove that the given set with the stated operati...
 4.1.21: In Exercises 1921, prove that the given set with the stated operati...
 4.1.22: Prove part (d) of Theorem 4.1.1.
 4.1.23: The argument that follows proves that if u, v, and w are vectors in...
 4.1.24: Let v be any vector in a vector space V. Prove that .
 4.1.25: Below is a sevenstep proof of part (b) of Theorem 4.1.1. Justify e...
 4.1.26: Let v be any vector in a vector space V. Prove that .
 4.1.27: Prove: If u is a vector in a vector space V and k a scalar such tha...
 4.1.a: n parts (a)(e) determine whether the statement is true or false, an...
 4.1.b: n parts (a)(e) determine whether the statement is true or false, an...
 4.1.c: n parts (a)(e) determine whether the statement is true or false, an...
 4.1.d: n parts (a)(e) determine whether the statement is true or false, an...
 4.1.e: n parts (a)(e) determine whether the statement is true or false, an...
Solutions for Chapter 4.1: Real Vector Spaces
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 4.1: Real Vector Spaces
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. Since 32 problems in chapter 4.1: Real Vector Spaces have been answered, more than 13809 students have viewed full stepbystep solutions from this chapter. Chapter 4.1: Real Vector Spaces includes 32 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.