 4.1.1: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.2: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.3: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.4: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.5: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.6: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.7: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.8: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.9: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.10: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.11: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.12: Questions 112 grow out of Figures 4.2 and 4.3 with four subspaces.
 4.1.13: Questions 1323 are about orthogonal subspaces.
 4.1.14: Questions 1323 are about orthogonal subspaces.
 4.1.15: Questions 1323 are about orthogonal subspaces.
 4.1.16: Questions 1323 are about orthogonal subspaces.
 4.1.17: Questions 1323 are about orthogonal subspaces.
 4.1.18: Questions 1323 are about orthogonal subspaces.
 4.1.19: Questions 1323 are about orthogonal subspaces.
 4.1.20: Questions 1323 are about orthogonal subspaces.
 4.1.21: Questions 1323 are about orthogonal subspaces.
 4.1.22: Questions 1323 are about orthogonal subspaces.
 4.1.23: Questions 1323 are about orthogonal subspaces.
 4.1.24: Questions 2430 are about perpendicular columns and rows.
 4.1.25: Questions 2430 are about perpendicular columns and rows.
 4.1.26: Questions 2430 are about perpendicular columns and rows.
 4.1.27: Questions 2430 are about perpendicular columns and rows.
 4.1.28: Questions 2430 are about perpendicular columns and rows.
 4.1.29: Questions 2430 are about perpendicular columns and rows.
 4.1.30: Questions 2430 are about perpendicular columns and rows.
 4.1.31: The command N = nulI(A) will produce a basis for the nullspace of A...
 4.1.32: Suppose I give you four nonzero vectors r, n, c, I in R 2 (a) What ...
 4.1.33: Suppose I give you eight vectors r I, r2, nl, n2, CI, C2, 11,12 in ...
Solutions for Chapter 4.1: Orthogonality of the Four Subspaces
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 4.1: Orthogonality of the Four Subspaces
Get Full SolutionsChapter 4.1: Orthogonality of the Four Subspaces includes 33 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. This expansive textbook survival guide covers the following chapters and their solutions. Since 33 problems in chapter 4.1: Orthogonality of the Four Subspaces have been answered, more than 11043 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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

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

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).

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.