 7.9.7.1.155: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.156: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.157: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.158: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.159: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.160: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.161: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.162: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.163: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.164: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.165: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.166: Is the given set (taken with the usual addition and scalar multipli...
 7.9.7.1.167: (Different bases) Find three bases for R2.
 7.9.7.1.168: (Uniqueness) Show that the representation v = c1a(1) + ... + cna(n)...
 7.9.7.1.169: Find the inverse transformation. (Show the details of your work.)
 7.9.7.1.170: Find the inverse transformation. (Show the details of your work.)
 7.9.7.1.171: Find the inverse transformation. (Show the details of your work.)
 7.9.7.1.172: Find the inverse transformation. (Show the details of your work.)
 7.9.7.1.173: Find the inverse transformation. (Show the details of your work.)
 7.9.7.1.174: Find the inverse transformation. (Show the details of your work.)
 7.9.7.1.175: Find the Euclidean nonn of the vectors [4 2 6]T
 7.9.7.1.176: Find the Euclidean nonn of the vectors [0 3 3 0 5 I]T
 7.9.7.1.177: Find the Euclidean nonn of the vectors[16 32 O]T
 7.9.7.1.178: Find the Euclidean nonn of the vectors [~ I ! 2f
 7.9.7.1.179: Find the Euclidean nonn of the vectors [0 1 0 0 1 l]T
 7.9.7.1.180: Find the Euclidean nonn of the vectors [~ ~ if
 7.9.7.1.181: (Orthogonality) Show that the vectors in Probs. 21 and 23 are ortho...
 7.9.7.1.182: Find all vectors v in R3 orthogonal to [2 0 I]T.
 7.9.7.1.183: (Unit vectors) Find all unit vectors orthogonal to [4 3]T. Make a ...
 7.9.7.1.184: (Triangle inequality) Verify (4) for the vectors in Probs. 21 and 23.
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
Get Full SolutionsChapter 7.9: Vector Spaces, Inner Product Spaces. Linear Transformations. Optional includes 30 full stepbystep 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 stepbystep solutions from this chapter.

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.

GramSchmidt 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 = Ql. 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 = Q1 = 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)jI 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).