 2.1: Let A = [ 2 7 O J.B = [ _f ]. c= [  J.andD=[6] 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
ISBN: 9781449679545
Solutions for Chapter 2: Matrices and Linear Transformations
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2: Matrices and Linear Transformations includes 34 full stepbystep solutions. Since 34 problems in chapter 2: Matrices and Linear Transformations have been answered, more than 8487 students have viewed full stepbystep solutions from this chapter.

Back substitution.
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).

Distributive Law
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.

Factorization
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 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Normal matrix.
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.

Schwarz inequality
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. AI 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 AI 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.