- 2.2.1: Suppose that T : R3 R2 is a linear transformation and that T 1 2 1 ...
- 2.2.2: Suppose that T : R3 R2 is defined by T x1 x2 x3 = _ x1 + 2x2 + x3 3...
- 2.2.3: Suppose T : R2 R2 is a linear transformation. In each case, use the...
- 2.2.4: Determine whether each of the following functions is a linear trans...
- 2.2.5: Give 2 2 matrices A so that for any x R2 we have, respectively: a. ...
- 2.2.6: Let T : R2 R2 be the linear transformation defined by rotating the ...
- 2.2.7: a. Calculate AA and AA . (Recall the definition of the rotation mat...
- 2.2.8: Let A be the rotation matrix defined on p. 98, 0 . Prove that a. _A...
- 2.2.9: Let _ be the line spanned by a R2, and let R_ : R2 R2 be the linear...
- 2.2.10: Let T : Rn Rm be a linear transformation. Prove the following: a. T...
- 2.2.11: a. Prove that if T : Rn Rm is a linear transformation and c is any ...
- 2.2.12: a. Let _ be the line spanned by _ cos sin . Show that the standard ...
- 2.2.13: Let _ be a line through the origin in R2. a. Show that P2 _ = P_ P_...
- 2.2.14: Let _1 be the line through the origin in R2 making angle with the x...
- 2.2.15: Let _ R2 be a line through the origin. a. Give a geometric argument...
Solutions for Chapter 2.2: Linear Transformations: An Introduction
Full solutions for Linear Algebra: A Geometric Approach | 2nd Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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 n-l - An).
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
Invert A by row operations on [A I] to reach [I A-I].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.