- 2.1: Let A = [ 2 7 O J.B = [ _f ]. c= [ - -J.andD= Compute the follow...
- 2.2: Let A be a 2 2 matrix, Ba 2 2 matrix, Ca 2 3 matrix, D a 3 2 matrix...
- 2.3: If A c [ 2 -4 7 1 determine the following element of D 2 AB - C wit...
- 2.4: a) Let A 2 and B 0 -1 Compute the product AB using the columns of B...
- 2.5: If A [ i] B . [ - ] and C compute each of the following. (a) (A1) 2...
- 2.6: Consider the followwg sysli!m of equations. You are given two solut...
- 2.7: Find the subspace 3 + + 3+ of solutions for the following homogeneo...
- 2.8: Consider the followin!1i nonhomogeneous system of linear equations....
- 2.9: Determine the inverse of each of the following matrices, if it exis...
- 2.10: Use the matrix inverse method to solve the following system of equa...
- 2.11: Find A such that A- [ 5 -6] . -2g. 3 4
- 2.12: Verify the associative property of multiplication A (BC) (AB )C
- 2.13: (a) LetT and T2 be the following row operations. T : interchange ro...
- 2.14: If n is a nonnegative integer and c is a scalar, prove that (c A)n ...
- 2.15: Let A be a matrix such that AA1 0 Show that A 0
- 2.16: A matrix is said to be normal if AA1 A1A Prove that all symmetric m...
- 2.17: Amatrix AisidempotentifA2 A Prove that ifAis idempotent then A 1 is...
- 2.18: A matrix .. is nilpotent if. _P _ for some positive integer p. The ...
- 2.19: Prove that if A is symmetric and invertible, then A - t is also sym...
- 2.20: Prove that a matrix with a row of zeros or a column of zeros has no...
- 2.21: Consider the transformation T defined by the following matrix A. De...
- 2.22: Determine whether the following transformations are linear. (a) T(x...
- 2.23: Find the matrix that maps: [ r:1 r aps R'-> R' such iliat  e> [ ]
- 2.24: Find a single matrix that defines a rotation of the plane through a...
- 2.25: Determine the matrix that defines a reflection about the line y = -x.
- 2.26: Determine the matrix that defines a projection onto the line y = -x.
- 2.27: Find theequationofthe image ofthe liney = -5x + 1 under a scaling o...
- 2.28: Find the equation of the image of the line y 2x + 3 under a shear o...
- 2.29: Construct a 2 X 2 matrix that defines a shear of factor 3 in the y-...
- 2.30: Compute . _ . _ and AB for the following matrices, and show that A ...
- 2.31: Prove that every real symmetric matrix is hermitian.
- 2.32: The following matrix A describes the pottery contents of various gr...
- 2.33: The following stochastic matrix P gives the probabilities for a cer...
- 2.34: Let A be the adjacency matrix of a digraph. What do you know about ...
Solutions for Chapter 2: Matrices and Linear Transformations
Full solutions for Linear Algebra with Applications | 8th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
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).
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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·
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.