 4.2.1: Refer to Exercise 1 of Section 1. For each linear transformation L,...
 4.2.2: For each of the following linear transformations L mapping R3 into ...
 4.2.3: For each of the following linear operators L on R3, find a matrix A...
 4.2.4: Let L be the linear operator on R3 defined by L(x) = 2x1 x2 x3 2x2 ...
 4.2.5: Find the standard matrix representation for each of the following l...
 4.2.6: Let b1 = 1 1 0 , b2 = 1 0 1 , b3 = 0 1 1 and let L be the linear tr...
 4.2.7: Let y1 = 1 1 1 , y2 = 1 1 0 , y3 = 1 0 0 and let I be the identity ...
 4.2.8: Let y1, y2, and y3 be defined as in Exercise 7, and let L be the li...
 4.2.9: Let R = 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 The column vectors of R repre...
 4.2.10: For each of the following linear operators on R2, find the matrix r...
 4.2.11: Determine the matrix representation of each of the following compos...
 4.2.12: Let Y , P, and R be the yaw, pitch, and roll matrices given in equa...
 4.2.13: Let L be the linear transformation mapping P2 into R2 defined by L(...
 4.2.14: The linear transformation L defined by L(p(x)) = p_ (x) + p(0) maps...
 4.2.15: Let S be the subspace of C[a, b] spanned by ex , xex , and x2ex . L...
 4.2.16: Let L be a linear operator on Rn. Suppose that L(x) = 0 for some x ...
 4.2.17: Let L be a linear operator on a vector space V. Let A be the matrix...
 4.2.18: Let E = {u1, u2, u3} and F = {b1, b2}, where u1 = 1 0 1 , u2 = 1 2 ...
 4.2.19: Suppose that L1 : V W and L2 : W Z are linear transformations and E...
 4.2.20: Let V and W be vector spaces with ordered bases E and F, respective...
Solutions for Chapter 4.2: Matrix Representations of Linear Transformations
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 4.2: Matrix Representations of Linear Transformations
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Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.