- Chapter 1: Systems of Linear Equations
- Chapter 2: Matrices
- Chapter 3: Determinants
- Chapter 4: Vector Spaces
- Chapter 5: Inner Product Spaces
- Chapter 6: Inner Product Spaces
- Chapter 7: Eigenvalues and Eigenvectors
Elementary Linear Algebra 7th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra | 7th Edition
ISBN: 9781133110873
Solutions by Chapter
Elementary Linear Algebra was written by and is associated to the ISBN: 9781133110873. The full step-by-step solution to problem in Elementary Linear Algebra were answered by , our top Math solution expert on 01/03/18, 08:36PM. Since problems from 7 chapters in Elementary Linear Algebra have been answered, more than 31719 students have viewed full step-by-step answer. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 7. This expansive textbook survival guide covers the following chapters: 7.
Key Math Terms and definitions covered in this textbook
-
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
-
Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
-
Column space C (A) =
space of all combinations of the columns of A.
-
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
-
Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
-
Condition number
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
-
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
-
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
-
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.
-
Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
-
Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and - ).
-
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)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).
-
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
-
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
-
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
-
Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
-
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.