 4.5.1E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
 4.5.2E: (a) find a basis, and (b) state the dimension.
 4.5.3E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
 4.5.4E: (a) find a basis, and (b) state the dimension.
 4.5.5E: (a) find a basis, and (b) state the dimension.
 4.5.6E: (a) find a basis, and (b) state the dimension.
 4.5.7E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
 4.5.8E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
 4.5.9E: Find the dimension of the subspace of all vectors in whose first an...
 4.5.10E: Find the dimension of the subspace H of ?2 spanned by
 4.5.11E: Find the dimension of the subspace spanned by the given vectors.
 4.5.12E: Find the dimension of the subspace spanned by the given vectors.
 4.5.13E: Determine the dimensions of Nul A and Col A for the matrices shown ...
 4.5.14E: Determine the dimensions of Nul A and Col A for the matrices.
 4.5.15E: Determine the dimensions of Nul A and Col A for the matrices.
 4.5.16E: Determine the dimensions of Nul A and Col A for the matrices.
 4.5.17E: Determine the dimensions of Nul A and Col A for the matrices.
 4.5.18E: Determine the dimensions of Nul A and Col A for the matrices.
 4.5.19E: In Exercises 19 and 20, V is a vector space. Mark each statement Tr...
 4.5.20E: In Exercises 19 and 20, V is a vector space. Mark each statement Tr...
 4.5.21E: The first four Hermite polynomials are and . These polynomials aris...
 4.5.22E: The first four Laguerre polynomials are , and . Show that these pol...
 4.5.23E: Let be the basis of ?3 consisting of the Hermite polynomials in Exe...
 4.5.24E: Let be the basis of ?2 consisting of the first three Laguerre polyn...
 4.5.25E: Let S be a subset of an ndimensional vector space V, and suppose S...
 4.5.26E: Let H be an ndimensional subspace of an ndimensional vector space...
 4.5.27E: Explain why the space of all polynomials is an infinitedimensional...
 4.5.28E: Show that the space of all continuous functions defined on the real...
 4.5.29E: In Exercises 29 and 30, V is a nonzero finitedimensional vector sp...
 4.5.30E: In Exercises 29 and 30, V is a nonzero finitedimensional vector sp...
 4.5.31E: Exercises 31 and 32 concern finitedimensional vector spaces V and ...
 4.5.32E: Exercises 31 and 32 concern finitedimensional vector spaces V and ...
 4.5.33E: [M] According to Theorem 11, a linearly independent set in can be e...
 4.5.34E: Assume the following trigonometric identities (see Exercise 37 in S...
Solutions for Chapter 4.5: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 4.5
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.5 includes 34 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Since 34 problems in chapter 4.5 have been answered, more than 40800 students have viewed full stepbystep solutions from this chapter.

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

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

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

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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 •

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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 ().

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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.

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.