×
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
×

Solutions for Chapter 7.9: Vector Spaces, Inner Product Spaces. Linear Transformations. Optional

Full solutions for Advanced Engineering Mathematics | 9th Edition

ISBN: 9780471488859

Solutions for Chapter 7.9: Vector Spaces, Inner Product Spaces. Linear Transformations. Optional

Solutions for Chapter 7.9
4 5 0 328 Reviews
24
4
ISBN: 9780471488859

Chapter 7.9: Vector Spaces, Inner Product Spaces. Linear Transformations. Optional includes 30 full step-by-step solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 30 problems in chapter 7.9: Vector Spaces, Inner Product Spaces. Linear Transformations. Optional have been answered, more than 48746 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Column space C (A) =

space of all combinations of the columns of A.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• 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.

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• Linear combination cv + d w or L C jV j.

Vector addition and scalar multiplication.

• Matrix multiplication AB.

The i, j entry of AB is (row i of A)ยท(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 .

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• 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.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Solvable system Ax = b.

The right side b is in the column space of A.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Standard basis for Rn.

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

• 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 v in Rn.

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

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

×