 4.2.1: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.2: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.3: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.4: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.5: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.6: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.7: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.8: In Exercises 1 8. find a basis for (a) the column space tmd (b) th...
 4.2.9: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.10: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.11: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.12: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.13: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.14: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.15: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.16: In Exercises 9 16. a linec1r transformation T is gien. (a) Find a ...
 4.2.17: In Exercises 1732. find a basis for each subspace. 17. {[_ 2 ;]en2...
 4.2.18: In Exercises 1732. find a basis for each subspace. 18. { [ ; ~~;]...
 4.2.19: In Exercises 1732. find a basis for each subspace. 19. ~S E R 4: ...
 4.2.20: In Exercises 1732. find a basis for each subspace. 20. 2r 5r+ 6s ...
 4.2.21: In Exercises 1732. find a basis for each subspace. 21. ~:] en3 :x,...
 4.2.22: In Exercises 1732. find a basis for each subspace. 22. {[;~] e R 3...
 4.2.23: In Exercises 1732. find a basis for each subspace. 23 j[~]R' .. ~...
 4.2.24: In Exercises 1732. find a basis for each subspace. 24. ml <R' <>+'...
 4.2.25: In Exercises 1732. find a basis for each subspace. 25. Span { GJ, ...
 4.2.26: In Exercises 1732. find a basis for each subspace. 26. Span { L:J....
 4.2.27: In Exercises 1732. find a basis for each subspace. 27. Span { [ ll
 4.2.28: In Exercises 1732. find a basis for each subspace. 28. Span { [ j]. [
 4.2.29: In Exercises 1732. find a basis for each subspace. 29 s,m l [!H
 4.2.30: In Exercises 1732. find a basis for each subspace. 30 Span JO. ,~. m
 4.2.31: In Exercises 1732. find a basis for each subspace. 31 s~ l
 4.2.32: In Exercises 1732. find a basis for each subspace. 32 Span, , . ., l
 4.2.33: Every nonzero subspace of R" has a unique basis.
 4.2.34: Every nonzero subspace o f R" has a basis
 4.2.35: A basis for a subspace is a generating sel thai is as large as poss...
 4.2.36: If Sis a linearly independent set and SpanS = \1, then S is a basis...
 4.2.37: Every finite generating set for a subspace contains a basis for the...
 4.2.38: A basis for a subspace is a linearly independent subset of the subs...
 4.2.39: Every basis for a particular subspace contains the same number of v...
 4.2.40: The columns of any matrix form a basis for its column space.
 4.2.41: The pivot columns of the reduced row echelon form of A form a basis...
 4.2.42: The vectors in the vector form of the general solution of Ax = 0 fm...
 4.2.43: If V is a subspace of dimension k. then every generati ng set for V...
 4.2.44: lf V is a subspace of dimension k, then every generating set for V ...
 4.2.45: If S is a linearly independent set of k vectors from a subspace V o...
 4.2.46: If V is a subspace of dimension k. then every set containing more t...
 4.2.47: The dimension of n is /1 ,
 4.2.48: The vectors in the standard basis for n arc the standard vectors of n.
 4.2.49: Every linearly independent subset of a subspace is contained in a b...
 4.2.50: Every subspace of n has a basis composed of standard vectors.
 4.2.51: A basis for the null space of a linear transformation is also a bas...
 4.2.52: A basis for the range of a linear transformation is also a basis fo...
 4.2.53: ~plO" why 1 [;] [ l] [ j]j ;, "~ g~o< ing set for n4
 4.2.54: Explain why { [ !]
 4.2.55: Explain why { [ ] , [ n } is not a basis ror n
 4.2.56: Explain why { [i] , [~] } is not a generating set for nJ
 4.2.57: Explain why {[_:J.[;J.[~J.[m is not linearly independent
 4.2.58: Explain why { [ _;] , [ 7]. [ _:]} is not a basis for n2
 4.2.59: Show that { [ ~ l [n } is a basis for the subspace in
 4.2.60: Show that :::'::.:'! [ _j]. [ ll l ;, ' oo,;, '" "' "'"'= ,, :,::~...
 4.2.61: Show that :,::~~ !t l] [ !]
 4.2.62: Show that ::::":" 1l1ifll Dll ;, ' ~;, fdo
 4.2.63: Show tjat :::::.:rrfnm ;, ' oo,;, '" ,,, """ .,
 4.2.64: Show that ~::: mn!Jl ;, bul '"' <ho '"" p
 4.2.65: Show that ~::: mn!Jl ;,
 4.2.66: Show hm 1 Ul [ ll l ;,
 4.2.67: What is the dimension of Span {v}. where v 'I 0? Justify your answer
 4.2.68: What is the dimension of the subspace 1
 4.2.69: What is the dimension of the subspace 1 [I] ' R" " = 0 '"' " = +"""'
 4.2.70: Find the dimension of the subspace Justify your answer.
 4.2.71: Let A= {u 1 u2 ..... Uk} be a basis for a k dimensional subspace V...
 4.2.72: Let A = {u t, u 2, . .. , Uk} be a basis for a kdimensional subspa...
 4.2.73: Let A= {UJ. u2 ..... Ut } be a basis for a k dimcnsional subspace ...
 4.2.74: Let A = {u 1 u2 . .. , ud be a basis for a k dime nsional subspace...
 4.2.75: Let T: R" > nm be a linear transformation and {u t. u 2 ..... u.,}...
 4.2.76: Let T: R" + 'R!" be a onetoone linear tran sformation and V be a...
 4.2.77: Let V and W be nonzero subspaces of R" such that each vector u in R...
 4.2.78: Let V be a subspace o f n. According to Theorem 4 .4, a linearly in...
 4.2.79: In Exercises 7982. use tile procedure described in Etercise 78 to ...
 4.2.80: In Exercises 7982. use tile procedure described in Etercise 78 to ...
 4.2.81: In Exercises 7982. use tile procedure described in Etercise 78 to ...
 4.2.82: In Exercises 7982. use tile procedure described in Etercise 78 to ...
 4.2.83: Let v = { [: ] e n3 : v,  v2 + "J = o} S = {[ll[!J.[!]} (a) Show...
 4.2.84: "'" ~ lml "'' ,,, ,, o '"'" ~ o ] " s~ l[llm [j]] (a) Show that S...
 4.2.85: Let [ 0.1 A = 0.7 0.5 0.2 0.3J 0.9 1.23 0.5 1.75 0.5  0.5 0.5 ...
 4.2.86: Show that [  57.1 29.0] [26.6] 53.8 16.0 . 7.0 4.9 9.1  7.0 13...
 4.2.87: Show that { [ 1.1] [  2.7] [ 2.5] } =7.8 . 7.6 . 5 9.0 4.0 6.5 is ...
 4.2.88: Let ~  0.21 0.2 0.58 0.4 A= 0.3 0.63  0.1  0.59 0.5 1.2 2.52 0....
Solutions for Chapter 4.2: SUBSPACES AND THEIR PROPERTIES
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 4.2: SUBSPACES AND THEIR PROPERTIES
Get Full SolutionsElementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. Chapter 4.2: SUBSPACES AND THEIR PROPERTIES includes 88 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Since 88 problems in chapter 4.2: SUBSPACES AND THEIR PROPERTIES have been answered, more than 25254 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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

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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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 .

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

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(D» O.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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