 5.2.1: Let T: R3 R2be a linear transformation defined as follows on the st...
 5.2.2: Let T: R2 R be a linear transformation defined as follows on the st...
 5.2.3: Let T be a linear operator on R2 defined as follows on the standard...
 5.2.4: Let T: P 2 P1 be a linear transformation defined as follows on the ...
 5.2.5: Let T be a linear operator on P 2 defined as follows on the standar...
 5.2.6: Let T: U V be a linear transformation. Let T be defined relative to...
 5.2.7: LetT: U V be a linear transformation. Let Tbe defined relative to b...
 5.2.8: Let T: U V be a linear transformation. Let T be defined relative to...
 5.2.9: Find the matrices of the following linear transformations of R 3 R ...
 5.2.10: Find the matrices of the following linear operators on R 3 with res...
 5.2.11: Consider the linear transformation T: R 3 R 2 defined by T(x, y, z)...
 5.2.12: Consider the linear transformation T: R 2 R 3 defined by T(x, y) = ...
 5.2.13: Consider the linear operator T: R 2 R 2 defined by T(x, y) = (2x, x...
 5.2.14: Find the matrix of the differential operator D with respect to the ...
 5.2.15: Let Vbe the vector space of functions having domain [ 0, 1T] genera...
 5.2.16: Find the matrix of the following linear transformations with respec...
 5.2.17: Find the matrix of the following linear transformation T of P2 into...
 5.2.18: Find the matrix of the following linear operator Ton P 1 with respe...
 5.2.19: Find the matrix of the following linear operator Ton P 1 with respe...
 5.2.20: Consider the linear operator T(x, y) = {2x, x + y) on R 2 Find the ...
 5.2.21: Consider the linear operator T ( x, y) = ( x y, x + y) on R 2 Find...
 5.2.22: a) Let V and Wbe vector spaces and let Ube a subspace of V. Is it a...
 5.2.23: Construct a linear transformation of R2 into R2 that has the subspa...
 5.2.24: Construct a linear transformation of R 2 into R 3 that has the subs...
 5.2.25: LetT: U Ufor a vectorspaceUbe defined byT{u) = u. Prove that Tis li...
 5.2.26: LetT: U U for a vector space Ube defined by T( u) = 0. Prove that T...
 5.2.27: Let U, V, and W be vector spaces with bases B = {ui. ... 'Un}, B' =...
 5.2.28: Is it possible for two distinct linear transformations T: U V and L...
Solutions for Chapter 5.2: Matrix Representations of Linear Transformations
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 5.2: Matrix Representations of Linear Transformations
Get Full SolutionsChapter 5.2: Matrix Representations of Linear Transformations includes 28 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. Since 28 problems in chapter 5.2: Matrix Representations of Linear Transformations have been answered, more than 8434 students have viewed full stepbystep solutions from this chapter.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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 .

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).

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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}.

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