 1.7.1: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.2: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.3: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.4: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.5: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.6: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.7: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.8: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.9: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.10: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.11: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.12: urcises 1 12, determine by inspection ll'hether the gilen sets are...
 1.7.13: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.14: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.15: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.16: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.17: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.18: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.19: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.20: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.21: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.22: In Exercises 13 22, a setS is given. Detennine by impection a subs...
 1.7.23: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.24: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.25: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.26: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.27: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.28: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.29: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.30: In Exe~i:s 23 30. d~tennine whether the given set is linearly mdepe...
 1.7.31: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.32: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.33: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.34: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.35: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.36: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.37: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.38: In Exe~ises 31 38, a linearly dependem setS is given. Write some v...
 1.7.39: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.40: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.41: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.42: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.43: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.44: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.45: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.46: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.47: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.48: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.49: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.50: In Exercises 39 50, derennine, if possible, a l'tliue of r for whi...
 1.7.51: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.52: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.53: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.54: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.55: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.56: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.57: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.58: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.59: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.60: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.61: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.62: In Exercises 51 62, wrire rhe vecror fonn of rile general .wlwion ...
 1.7.63: If Sis linearly independent, then no vector inS is a lineM combi na...
 1.7.64: If the onl y solution of Ax = 0 is 0. then the rows of A are linear...
 1.7.65: If the nullity of A is 0, then the columns of A are linearly depend...
 1.7.66: If the columns of the reduced row echelon form of A are distinct st...
 1.7.67: If A is an 111 x 11 matrix with rank 11, then the columns of A are ...
 1.7.68: A homogeneous equation i> alway consistent.
 1.7.69: A homogeneous equation always has infinitely many solulion~.
 1.7.70: If a vector form of the general ;olution of Ax= 0 is obtained by th...
 1.7.71: For any vector v. {vi is linearly dependent.
 1.7.72: A set of vectors in R!' is linearly dependent if and only if one of...
 1.7.73: lf a subset of n is linearly dependent. then it must contain at lea...
 1.7.74: If the columns of a 3 x .f matrix arc distinct. then they are linea...
 1.7.75: For the ;ystcm of linear equation; Ax = b to be homogeneous, b must...
 1.7.76: f a subset of n contains more than 11 vector., then it is linearly ...
 1.7.77: n the matrix equation Ax = b is consistent for every b in
 1.7.78: If every row of an m x 11 matrix A contains a pivot position. then ...
 1.7.79: . 1f <'o llo+czuz+ .. +ctu.t= O for <'t=n= .. = <'t = 0. then {uo. ...
 1.7.80: Any >Ubset of n that contain' 0 i\ hncarly dependent
 1.7.81: The sCI of Standard vector' in n i' linearly independent
 1.7.82: The largest number of linearly independent vectors in n is 11.
 1.7.83: Find a 2 x 2 matrix A such that () is the only solution of Ax = ()
 1.7.84: Find a 2 x 2 matrix A such that Ax = 0 has infinitely many solutions
 1.7.85: Find an example of linearly independent sub>cts {u 0, u2} and {\'1 ...
 1.7.86: Let (u1 uz ..... u.t} be a linearly independent set of vectors in n...
 1.7.87: Let u :ond v be distinct vecton< in R". Prove that the set {u. v} i...
 1.7.88: Let u. v. and w be distinct vectors in R". Prove that (u. v. wl is ...
 1.7.89: Pro'c that if (u o. uz, .... ud i~ a linearly independent 'ub>et of...
 1.7.90: Complete the proof of Theorem 1.9 by showing that if u o = 0 or u, ...
 1.7.91: Prove th:ot :ony nonempty subset of a linearly independent subset o...
 1.7.92: Prove th:ot if S o is a linearly dependent subset ofR" that is cont...
 1.7.93: Let S = {u0 111 ..... Ut} be a nonempty set of veeton< from n. Prov...
 1.7.94: State and prove the con\'crsc of Exercise 93.
 1.7.95: LetS = {u ,. uz ..... Ut } be a noncmpty sub>ct of R. and A be an m...
 1.7.96: Give an example to show that the preceding exercise is false if lil...
 1.7.97: etS= (u1 u2 .. lit} be a nonempty subset of 'R" and A be an m x 11 ...
 1.7.98: Let A and 8 be m x 11 matrices :.uch that 8 can be obtained by perf...
 1.7.99: Prove that if u matrix is in reduced row echelon form. then its non...
 1.7.100: Prove that the rows of an m x 11 matrix A :ore linearly independent...
 1.7.101: In terl'ises /OJ 104. use t'ither a calculmor ll'ith matri.l capabi...
 1.7.102: In terl'ises /OJ 104. use t'ither a calculmor ll'ith matri.l capabi...
 1.7.103: In terl'ises /OJ 104. use t'ither a calculmor ll'ith matri.l capabi...
 1.7.104: In terl'ises /OJ 104. use t'ither a calculmor ll'ith matri.l capabi...
Solutions for Chapter 1.7: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 1.7: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS
Get Full SolutionsElementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. Since 104 problems in chapter 1.7: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS have been answered, more than 21453 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.7: MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS includes 104 full stepbystep solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

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

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

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 •

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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