 7.3.1: For Exercises 16, determine by inspection if the set V could possib...
 7.3.2: For Exercises 16, determine by inspection if the set V could possib...
 7.3.3: For Exercises 16, determine by inspection if the set V could possib...
 7.3.4: For Exercises 16, determine by inspection if the set V could possib...
 7.3.5: For Exercises 16, determine by inspection if the set V could possib...
 7.3.6: For Exercises 16, determine by inspection if the set V could possib...
 7.3.7: For Exercises 712, determine if the set V is a basis for V.V = {2x2...
 7.3.8: For Exercises 712, determine if the set V is a basis for V.V = {x2 ...
 7.3.9: For Exercises 712, determine if the set V is a basis for V.V =1 22 ...
 7.3.10: For Exercises 712, determine if the set V is a basis for V.V =4 32 ...
 7.3.11: For Exercises 712, determine if the set V is a basis for V.V = {(1,...
 7.3.12: For Exercises 712, determine if the set V is a basis for V.V = {1, ...
 7.3.13: For Exercises 1318, find a basis for the subspace S and determinedi...
 7.3.14: For Exercises 1318, find a basis for the subspace S and determinedi...
 7.3.15: For Exercises 1318, find a basis for the subspace S and determinedi...
 7.3.16: For Exercises 1318, find a basis for the subspace S and determinedi...
 7.3.17: For Exercises 1318, find a basis for the subspace S and determinedi...
 7.3.18: For Exercises 1318, find a basis for the subspace S and determinedi...
 7.3.19: For Exercises 1920, determine the dimension of the subspace S.Justi...
 7.3.20: For Exercises 1920, determine the dimension of the subspace S.Justi...
 7.3.21: For Exercises 2124, extend the linearly independent set V to abasis...
 7.3.22: For Exercises 2124, extend the linearly independent set V to abasis...
 7.3.23: For Exercises 2124, extend the linearly independent set V to abasis...
 7.3.24: For Exercises 2124, extend the linearly independent set V to abasis...
 7.3.25: For Exercises 2526, remove vectors from V to yield a basis for V.V ...
 7.3.26: For Exercises 2526, remove vectors from V to yield a basis for V.V ...
 7.3.27: C Exercises 2730 assume some knowledge of calculusLet S be the subs...
 7.3.28: C Exercises 2730 assume some knowledge of calculusFind a basis for ...
 7.3.29: C Exercises 2730 assume some knowledge of calculusFind a basis for ...
 7.3.30: C Exercises 2730 assume some knowledge of calculusDetermine the dim...
 7.3.31: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 7.3.32: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 7.3.33: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 7.3.34: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 7.3.35: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 7.3.36: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 7.3.37: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.38: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.39: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.40: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.41: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.42: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.43: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.44: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.45: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.46: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 7.3.47: Prove that if {v1, v2, ... , vk } is a basis for a vector space V,t...
 7.3.48: Prove that dim(R) =
 7.3.49: Show that1 00 0,0 10 0,0 01 0,0 00 1is a basis for R22.
 7.3.50: Give a basis for Rnm, and justify that your set forms a basis.Prove...
 7.3.51: If V is a vector space with dim(V) = m, prove that there existsubsp...
 7.3.52: Prove that dim(T(m, n)) = mn. (HINT: Recall that ifT : Rm Rn is a l...
 7.3.53: Prove Theorem 7.11: The set{v1, ... , vm}is a basis for a vectorspa...
 7.3.54: Prove Theorem 7.12: Suppose that V1 and V2 are both basesof a vecto...
 7.3.55: Let V1 and V2 be vector spaces with V1 a subset of V2.(a) If dim(V2...
 7.3.56: Prove Theorem 7.14: Let V = {v1, ... , vm} be a subset of afinite d...
 7.3.57: Prove Theorem 7.16: Let V = {v1, ... , vm} be a subset of avector s...
Solutions for Chapter 7.3: Basis and Dimension
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 7.3: Basis and Dimension
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Since 57 problems in chapter 7.3: Basis and Dimension have been answered, more than 15081 students have viewed full stepbystep solutions from this chapter. Chapter 7.3: Basis and Dimension includes 57 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

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).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Outer product uv T
= column times row = rank one matrix.

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

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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 ().

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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.