- 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 Change-of-Basis 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
Solutions by Chapter
This 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 step-by-step 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 34786 students have viewed full step-by-step answer.
Key Math Terms and definitions covered in this textbook
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
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Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
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Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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Gram-Schmidt 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.
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Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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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.
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Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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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.
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.
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Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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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.
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.