 3.1: If v1, v2,..., vn and w 1, w 2,..., w m are any two bases of a subs...
 3.2: If A is a 5 6 matrix of rank 4, then the nullity of A is 1.
 3.3: The image of a 3 4 matrix is a subspace of R4.
 3.4: The span of vectors v1, v2,..., vn consists of all linear combinati...
 3.5: If v1, v2,..., vn are linearly independent vectors in Rn, then they...
 3.6: There exists a 5 4 matrix whose image consists of all of R5.
 3.7: The kernel of any invertible matrix consists of the zero vector only.
 3.8: The identity matrix In is similar to all invertible n n matrices.
 3.9: If 2u + 3v + 4w = 5u + 6v + 7w , then vectors u, v, w must be linea...
 3.10: The column vectors of a 5 4 matrix must be linearly dependent
 3.11: If matrix A is similar to matrix B, and B is similar to C, then C m...
 3.12: If a subspace V of Rn contains none of the standard vectors e1, e2,...
 3.13: If vectors v1, v2, v3, v4 are linearly independent, then vectors v1...
 3.14: The vectors of the form a b 0 a (where a and b are arbitrary real n...
 3.15: Matrix 1 0 0 1 is similar to 0 1 1 0 .
 3.16: Vectors 1 0 0 , 2 1 0 , 3 2 1 form a basis of R3.
 3.17: If the kernel of a matrix A consists of the zero vector only, then ...
 3.18: If the image of an n n matrix A is all of Rn, then A must be invert...
 3.19: If vectors v1, v2,..., vn span R4, then n must be equal to 4.
 3.20: If vectors u, v, and w are in a subspace V of Rn, then vector 2u 3v...
 3.21: If A and B are invertible n n matrices, then AB must be similar to ...
 3.22: If A is an invertible n n matrix, then the kernels of A and A1 must...
 3.23: Matrix 0 1 0 0 is similar to 0 0 0 1 .
 3.24: Vectors 1 2 3 4 , 5 6 7 8 , 9 8 7 6 , 5 4 3 2 , 1 0 1 2 are linearly
 3.25: If a subspace V of R3 contains the standard vectors e1, e2, e3, the...
 3.26: If a 2 2 matrix P represents the orthogonal projection onto a line ...
 3.27: If A and B are n n matrices, and vector v is in the kernel of both ...
 3.28: If v1, v2, v3 are any three distinct vectors in R3, then there must...
 3.29: If v1, v2, v3 are any three distinct vectors in R3, then there must...
 3.30: If vectors u, v, w are linearly dependent, then vector w must be a ...
 3.31: R2 is a subspace of R3.
 3.32: If an nn matrix A is similar to matrix B, then A+7In must be simila...
 3.33: If V is any threedimensional subspace of R5, then V has infinitely...
 3.34: Matrix In is similar to 2In.
 3.35: If AB = 0 for two 2 2 matrices A and B, then B A must be the zero m...
 3.36: If A and B are n n matrices, and vector v is in the image of both A...
 3.37: If V and W are subspaces of Rn, then their union V W must be a subs...
 3.38: If the kernel of a 5 4 matrix A consists of the zero vector only an...
 3.39: If v1, v2,..., vn and w 1, w 2,..., w n are two bases of Rn, then t...
 3.40: If matrix A represents a rotation through /2 and matrix B a rotatio...
 3.41: There exists a 22 matrix A such that im(A) = ker(A).
 3.42: If two n n matrices A and B have the same rank, then they must be s...
 3.43: If A is similar to B, and A is invertible, then B must be invertibl...
 3.44: If A2 = 0 for a 10 10 matrix A, then the inequality rank(A) 5 must ...
 3.45: For every subspace V of R3, there exists a 3 3 matrix A such that V...
 3.46: There exists a nonzero 2 2 matrix A that is similar to 2A.
 3.47: . If the 2 2 matrix R represents the reflection about a line in R2,...
 3.48: If A is similar to B, then there exists one and only one invertible...
 3.49: If the kernel of a 5 4 matrix A consists of the zero vector alone, ...
 3.50: If A is any n n matrix such that A2 = A, then the image of A and th...
 3.51: There exists a 2 2 matrix A such that A2 = 0 and A3 = 0.
 3.52: If A and B are n m matrices such that the image of A is a subset of...
 3.53: Among the 3 3 matrices whose entries are all 0s and 1s, most are in...
Solutions for Chapter 3: Linear Algebra with Applications 5th Edition
Full solutions for Linear Algebra with Applications  5th Edition
ISBN: 9780321796974
Solutions for Chapter 3
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 3 includes 53 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321796974. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 5. Since 53 problems in chapter 3 have been answered, more than 3940 students have viewed full stepbystep solutions from this chapter.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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.

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 •

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

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.

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).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·