 4.2.1: In Exercises 14, determine if the vectors shown form a basis forR2....
 4.2.2: In Exercises 14, determine if the vectors shown form a basis forR2....
 4.2.3: In Exercises 14, determine if the vectors shown form a basis forR2....
 4.2.4: In Exercises 14, determine if the vectors shown form a basis forR2....
 4.2.5: In Exercises 510, use the solution method from Example 1 to finda b...
 4.2.6: In Exercises 510, use the solution method from Example 1 to finda b...
 4.2.7: In Exercises 510, use the solution method from Example 1 to finda b...
 4.2.8: In Exercises 510, use the solution method from Example 1 to finda b...
 4.2.9: In Exercises 510, use the solution method from Example 1 to finda b...
 4.2.10: In Exercises 510, use the solution method from Example 1 to finda b...
 4.2.11: In Exercises 1116, use the solution method from Example 2 tofind a ...
 4.2.12: In Exercises 1116, use the solution method from Example 2 tofind a ...
 4.2.13: In Exercises 1116, use the solution method from Example 2 tofind a ...
 4.2.14: In Exercises 1116, use the solution method from Example 2 tofind a ...
 4.2.15: In Exercises 1116, use the solution method from Example 2 tofind a ...
 4.2.16: In Exercises 1116, use the solution method from Example 2 tofind a ...
 4.2.17: In Exercises 1722, find a basis for the given subspace by deletingl...
 4.2.18: In Exercises 1722, find a basis for the given subspace by deletingl...
 4.2.19: In Exercises 1722, find a basis for the given subspace by deletingl...
 4.2.20: In Exercises 1722, find a basis for the given subspace by deletingl...
 4.2.21: In Exercises 1722, find a basis for the given subspace by deletingl...
 4.2.22: In Exercises 1722, find a basis for the given subspace by deletingl...
 4.2.23: In Exercises 2324, expand the given set to form a basis for R2. 13
 4.2.24: In Exercises 2324, expand the given set to form a basis for R2.04
 4.2.25: In Exercises 2528, expand the given set to form a basis for R3.121
 4.2.26: In Exercises 2528, expand the given set to form a basis for R3.105
 4.2.27: In Exercises 2528, expand the given set to form a basis for R3. 132...
 4.2.28: In Exercises 2528, expand the given set to form a basis for R3.213,326
 4.2.29: In Exercises 2932, find a basis for the null space of the givenmatr...
 4.2.30: In Exercises 2932, find a basis for the null space of the givenmatr...
 4.2.31: In Exercises 2932, find a basis for the null space of the givenmatr...
 4.2.32: In Exercises 2932, find a basis for the null space of the givenmatr...
 4.2.33: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.34: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.35: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.36: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.37: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.38: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.39: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.40: FIND AN EXAMPLE For Exercises 3340, find an example thatmeets the g...
 4.2.41: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.42: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.43: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.44: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.45: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.46: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.47: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.48: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.49: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.50: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.51: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.52: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.53: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.54: TRUE OR FALSE For Exercises 4154, determine if the statementis true...
 4.2.55: Suppose that S1 and S2 are nonzero subspaces, with S1 containedinsi...
 4.2.56: Suppose that S1 and S2 are nonzero subspaces, with S1 containedinsi...
 4.2.57: Show that the only subspace of Rn that has dimension n is Rn.
 4.2.58: Explain why Rn (n > 1) has infinitely many subspaces ofdimension 1.
 4.2.59: Prove the converse of Theorem 4.9: If every vector s of a subspaceS...
 4.2.60: Complete the proof of Theorem 4.15: Let U = {u1, ... , um}be a set ...
 4.2.61: Prove Theorem 4.16: Suppose that S1 and S2 are both subspacesof Rn,...
 4.2.62: Prove Theorem 4.17: Let U = {u1, ... , um} be a set of vectorsin a ...
 4.2.63: Suppose that a matrix A is in echelon form. Prove that thenonzero r...
 4.2.64: If the set {u1, u2, u3} spans R3 andA = u1 u2 u3,what is nullity(A)?
 4.2.65: Suppose that S1 and S2 are subspaces of Rn, with dim(S1) =m1 and di...
 4.2.66: Prove Theorem 4.10: Let A and B be equivalent matrices.Then the sub...
 4.2.67: Prove Theorem 4.11: Suppose that U = u1 umandV = v1 vmare two equiv...
 4.2.68: Give a general proof of Theorem 4.12: If S is a subspace of Rn,then...
 4.2.69: C In Exercises 6970, determine if the given set of vectors is abasi...
 4.2.70: C In Exercises 6970, determine if the given set of vectors is abasi...
 4.2.71: C In Exercises 7172, determine if the given set of vectors is abasi...
 4.2.72: C In Exercises 7172, determine if the given set of vectors is abasi...
 4.2.73: C In Exercises 7374, determine if the given set of vectors is abasi...
 4.2.74: C In Exercises 7374, determine if the given set of vectors is abasi...
Solutions for Chapter 4.2: Basis and Dimension
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 4.2: Basis and Dimension
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Chapter 4.2: Basis and Dimension includes 74 full stepbystep solutions. Since 74 problems in chapter 4.2: Basis and Dimension have been answered, more than 15060 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

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.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).