 Chapter 1.1: Vectors
 Chapter 1.2: Dot Product
 Chapter 1.3: Hyperplanes in Rn
 Chapter 1.4: Systems of Linear Equations and Gaussian Elimination
 Chapter 1.5: The Theory of Linear Systems
 Chapter 1.6: Some Applications
 Chapter 2.1: Matrix Operations
 Chapter 2.2: Linear Transformations: An Introduction
 Chapter 2.3: Inverse Matrices
 Chapter 2.4: Elementary Matrices: Rows Get Equal Time
 Chapter 2.5: The Transpose
 Chapter 3.1: Subspaces of Rn
 Chapter 3.2: The Four Fundamental Subspaces
 Chapter 3.3: Linear Independence and Basis
 Chapter 3.4: Dimension and Its Consequences
 Chapter 3.5: A Graphic Example
 Chapter 3.6: AbstractVector Spaces
 Chapter 4.1: Inconsistent Systems and Projection
 Chapter 4.2: Orthogonal Bases
 Chapter 4.3: The Matrix of a Linear Transformation and the ChangeofBasis Formula
 Chapter 4.4: Linear Transformations on Abstract Vector Spaces
 Chapter 5.1: Properties of Determinants
 Chapter 5.2: Cofactors and Cramers Rule
 Chapter 5.3: Signed Area in R2 and SignedVolume in R3
 Chapter 6.1: The Characteristic Polynomial
 Chapter 6.2: Diagonalizability
 Chapter 6.3: Applications
 Chapter 6.4: The Spectral Theorem
 Chapter 7.1: Complex Eigenvalues and Jordan Canonical Form
 Chapter 7.2: Computer Graphics and Geometry
 Chapter 7.3: Matrix Exponentials and Differential Equations
Linear Algebra: A Geometric Approach 2nd Edition  Solutions by Chapter
Full solutions for Linear Algebra: A Geometric Approach  2nd Edition
ISBN: 9781429215213
Linear Algebra: A Geometric Approach  2nd Edition  Solutions by Chapter
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra: A Geometric Approach, edition: 2. This expansive textbook survival guide covers the following chapters: 31. The full stepbystep solution to problem in Linear Algebra: A Geometric Approach were answered by , our top Math solution expert on 03/15/18, 05:30PM. Linear Algebra: A Geometric Approach was written by and is associated to the ISBN: 9781429215213. Since problems from 31 chapters in Linear Algebra: A Geometric Approach have been answered, more than 1494 students have viewed full stepbystep answer.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

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.

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

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

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).
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