Solutions by ChapterTextbook: Elementary Linear Algebra
Edition: 8Author: Ron Larson ISBN: 9781305658004
Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Since problems from 45 chapters in Elementary Linear Algebra have been answered, more than 263182 students have viewed full step-by-step answer. This expansive textbook survival guide covers the following chapters: 45. The full step-by-step solution to problem in Elementary Linear Algebra were answered by , our top Math solution expert on 01/12/18, 03:19PM.
- 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.
- 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.
- Dimension of vector space
dim(V) = number of vectors in any basis for V.
- 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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
- Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
- lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
- 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.
- Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
- 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.
- Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
- Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
- Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
- Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
- Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
- Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.
-
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
- Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
- Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
- Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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Ron Larson
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Chapters
1
Systems of Linear Equations
3 sections
206 questions
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2
Matrices
6 sections
397 questions
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3
Determinants
4 sections
272 questions
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4
Vector Spaces
8 sections
557 questions
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5
Inner Product Spaces
5 sections
400 questions
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6
Linear Transformations
5 sections
326 questions
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7
Eigenvalues and Eigenvectors
6 sections
260 questions
+39 more