 1.3.2: Determine the transpose of each of the matrices that follow. In add...
 1.3.3: Prove that (aA + bB)1 = aA* + bB1 for any A, B Mmxn ( F) and any a,...
 1.3.4: Prove that (A1 )* = A for each A MmXn (F).
 1.3.5: Prove that A + A1 is symmetric for any square matrix A.
 1.3.6: Prove that tr(aA + bB) = atr(A) + 6tr() for any A. B Mnxn (F).
 1.3.7: Prove that diagonal matrices are symmetric matrices.
 1.3.8: Determine whether the following sets are subspaces of R3 under the ...
 1.3.9: Let Wi, W3, and VV4 be as in Exercise 8. Describe Wi n W3, Wi n W4,...
 1.3.10: Prove that W, = {(oi,a 2 ,..., a n ) F" : ai + o2 + + a n = 0} is a...
 1.3.11: Is the set. W = {f(x) P(F): f(x) = 0 or f(x) has degree; n] a subsp...
 1.3.12: An m x n matrix A is called upper triangular if all entries lying b...
 1.3.13: Let S be a nonempty set and F a. field. Prove that for any SQ S, {/...
 1.3.14: Let S be a nonempty set and F a. field. Let C(S, F) denote the set ...
 1.3.15: Is the set of all differentiable realvalued functions defined on R...
 1.3.16: Let Cn (R.) denote the set of all realvalued functions defined on ...
 1.3.17: Let Cn (R.) denote the set of all realvalued functions defined on ...
 1.3.18: Prove that a subset W of a vector space; V is a subspace of V if an...
 1.3.19: Let W, and W2 be subspaces of a vector space V. Prove that W U W2 ...
 1.3.20: Prove that if W is a subspace of a vector space V and w\, w2, , wn ...
 1.3.21: Show that the set of convergent sequences {on} (i.e., those for whi...
 1.3.22: Let F\ and F2 be fiedds. A function g T(Fl ,F2) is called an even f...
 1.3.23: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.24: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.25: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.26: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.27: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.28: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.29: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.30: The following definitions are used in Exercises 23 30. Definition. ...
 1.3.31: Let W be a subspace of a vector space V over a field F. For any v V...
Solutions for Chapter 1.3: Subspaces
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 1.3: Subspaces
Get Full SolutionsLinear Algebra was written by and is associated to the ISBN: 9780130084514. Since 30 problems in chapter 1.3: Subspaces have been answered, more than 10930 students have viewed full stepbystep solutions from this chapter. Chapter 1.3: Subspaces includes 30 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 , edition: 4.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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 ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.