## Linear Algebra with Applications

**ABOUT THIS TEXTBOOK**

*Linear Algebra with Applications* by W. Keith Nicholson, traditionally published for many years is now being released as an open educational resource and part of Lyryx with Open Texts! Supporting today’s students and instructors requires much more than a textbook, which is why Dr. Nicholson opted to work with Lyryx Learning.

Overall, the aim of the textbook is to achieve a balance among computational skills, theory, and applications of linear algebra. It is a relatively advanced introduction to the ideas and techniques of linear algebra targeted for science and engineering students who need to understand not only how to use these methods but also gain insight into why they work.

The contents have enough flexibility to present a traditional introduction to the subject, or to allow for a more applied course. Chapters 1–4 contain a one-semester course for beginners whereas Chapters 5–9 contain a second semester course.

**Block matrix.**

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

**Cayley-Hamilton Theorem.**

peA) = det(A - AI) has peA) = zero matrix.

**Companion matrix.**

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).

**Covariance matrix:E.**

When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

**Cyclic shift**

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

**Diagonalizable matrix A.**

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

**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.

**Ellipse (or ellipsoid) x T Ax = 1.**

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

**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.

**Gauss-Jordan method.**

Invert A by row operations on [A I] to reach [I A-I].

**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.

**Jordan form 1 = M- 1 AM.**

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

**Left nullspace N (AT).**

Nullspace of AT = "left nullspace" of A because y T A = OT.

**Linear transformation T.**

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

**Pascal matrix**

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).

**Reduced row echelon form R = rref(A).**

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

**Right inverse A+.**

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

**Saddle point of I(x}, ... ,xn ).**

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

**Singular matrix A.**

A square matrix that has no inverse: det(A) = o.

**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·

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#### Answer

#### Work Step by Step

First, change $-2x_{1}-7x_{2}=-5$ to $2x_{1}+7x_{2}=5$ by multiplying both sides by negative 1. $-1*(-2x_{1}-7x_{2})=-1*(-5)$ $2x_{1}+7x_{2}=5$ Next, multiply the first equation by two on both sides to get a common $2x_{1}$ term in both equations, $2*(x_{1}+5x_{2})=2*(7)$ $2x_{1}+10x_{2}=14$ Now you can subtract one equation from the other to get a new equation with ONLY ONE TERM. $2x_{1}+10x_{2}=14$ -$2x_{1}+7x_{2}=5$ The $2x_{1}$ cancels out and you are left with $3x_{2}=9$ Divide both sides by 3 and receive $x_{2}=3$ Now you can plug this into any original equation to receive the answer. $x_{1}+5x_{2}=7$ $x_{1}+5(3)=7$ $x_{1}+15=7$ $x_{1}=7-15=-8$ Check these two values in the other equations to make sure of your answers. $-2x_{1}-7x_{2}=-5$ $-2(-8)-7(3)=-5$ $16-21=-5$ $-5=-5$ These values make sense, and are the answers. Basically, for these types of problems, the idea is to manipulate the equations to find one variable, and plug that variable in to find the other.

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## Answers with applications linear algebra

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