 5.2.1: For each of the following matrices, determine a basis for each of t...
 5.2.2: Let S be the subspace of R3 spanned by x = (1,1, 1)T . (a) Find a b...
 5.2.3: (a) Let S be the subspace of R3 spanned by the vectors x = (x1, x2,...
 5.2.4: Let S be the subspace of R4 spanned by x1 = (1, 0,2, 1)T and x2 = (...
 5.2.5: Let A be a 3 2 matrix with rank 2. Give geometric descriptions of R...
 5.2.6: Is it possible for a matrix to have the vector (3, 1, 2) in its row...
 5.2.7: Let aj be a nonzero column vector of an m n matrix A. Is it possibl...
 5.2.8: Let S be the subspace of Rn spanned by the vectors x1, x2, . . . , ...
 5.2.9: If A is an m n matrix of rank r , what are the dimensions of N(A) a...
 5.2.10: Prove Corollary 5.2.5. 1
 5.2.11: Prove: If A is an m n matrix and x Rn, then either Ax = 0 or there ...
 5.2.12: Let A be an m n matrix. Explain why the following are true: (a) Any...
 5.2.13: Let A be an m n matrix. Show that (a) if x N(ATA), then Ax is in bo...
 5.2.14: Let A be an m n matrix, B an n r matrix, and C = AB. Show that (a) ...
 5.2.15: Let U and V be subspaces of a vector space W. Show that if W = U V,...
 5.2.16: Let A be an m n matrix of rank r and let {x1, . . . , xr } be a bas...
 5.2.17: Let x and y be linearly independent vectors in Rn and let S = Span(...
Solutions for Chapter 5.2: Orthogonal Subspaces
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 5.2: Orthogonal Subspaces
Get Full SolutionsSince 17 problems in chapter 5.2: Orthogonal Subspaces have been answered, more than 7436 students have viewed full stepbystep solutions from this chapter. Chapter 5.2: Orthogonal Subspaces includes 17 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This expansive textbook survival guide covers the following chapters and their 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.

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).

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

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).

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 •

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.