 4.1.1: Which of the subsets of Pi given in Exercises 1 through 5 are subsp...
 4.1.2: Which of the subsets of Pi given in Exercises 1 through 5 are subsp...
 4.1.3: Which of the subsets of Pi given in Exercises 1 through 5 are subsp...
 4.1.4: Which of the subsets of Pi given in Exercises 1 through 5 are subsp...
 4.1.5: Which of the subsets of Pi given in Exercises 1 through 5 are subsp...
 4.1.6: Which of the subsets of M3x3 given in Exercises 6 through 11 are su...
 4.1.7: Which of the subsets of M3x3 given in Exercises 6 through 11 are su...
 4.1.8: Which of the subsets of M3x3 given in Exercises 6 through 11 are su...
 4.1.9: Which of the subsets of M3x3 given in Exercises 6 through 11 are su...
 4.1.10: Which of the subsets of M3x3 given in Exercises 6 through 11 are su...
 4.1.11: Which of the subsets of M3x3 given in Exercises 6 through 11 are su...
 4.1.12: Let V be the space of all infinite sequences of real numbers. (See ...
 4.1.13: Let V be the space of all infinite sequences of real numbers. (See ...
 4.1.14: Let V be the space of all infinite sequences of real numbers. (See ...
 4.1.15: Let V be the space of all infinite sequences of real numbers. (See ...
 4.1.16: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.17: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.18: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.19: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.20: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.21: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.22: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.23: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.24: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.25: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.26: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.27: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.28: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.29: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.30: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.31: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.32: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.33: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.34: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.35: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.36: Find a basis for each of the spaces in Exercises 16 through 9 and d...
 4.1.37: If B is a diagonal 3 x 3 matrix, what are the possible dimensions o...
 4.1.38: If B is a diagonal 4 x 4 matrix, what are the possible dimensions o...
 4.1.39: What is the dimension of the space of all upper triangular nxn matr...
 4.1.40: If u is any vector in R", what are the possible dimensions of the s...
 4.1.41: If B is any 3x3 matrix, what are the possible dimensions of the spa...
 4.1.42: If B is any nxn matrix, what are the possible dimensions of the spa...
 4.1.43: If matrix A represents the reflection about a line L in R2, what is...
 4.1.44: If matrix A represents the orthogonal projection onto a plane V in ...
 4.1.45: Find a basis of the space V of ail 3 x 3 matrices A that commute wi...
 4.1.46: In the linear space of infinite sequences, consider the subspace W ...
 4.1.47: A function /(/) from R to R is called even if f(t) = /(r), for all...
 4.1.48: Find a basis of each of the following linear spaces, and thus deter...
 4.1.49: Let L(Rm, R) be the set of all linear transformations from Rm to Rn...
 4.1.50: Find all the solutions of the differential equation f"(x) + 8 / ' (...
 4.1.51: Find all the solutions of the differential equation / " ( * )  7 /...
 4.1.52: Make up a secondorder linear DE whose solution space is spanned by...
 4.1.53: Show that in an ndimensional linear space we can find at most n li...
 4.1.54: Show that if W is a subspace of an ndimensional linear space V, th...
 4.1.55: Show that the space F(R, R) of all functions from R to R is infinit...
 4.1.56: Show that the space of infinite sequences of real numbers is infini...
 4.1.57: We say that a linear space V is finitely generated if it can be spa...
 4.1.58: In this exercise we will show that the functions cos(*) and sin(jc)...
Solutions for Chapter 4.1: Introduction to Linear Spaces
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 4.1: Introduction to Linear Spaces
Get Full SolutionsChapter 4.1: Introduction to Linear Spaces includes 58 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Since 58 problems in chapter 4.1: Introduction to Linear Spaces have been answered, more than 15494 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

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

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).

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > 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 .

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.