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Chapter 1: Q70E (page 39)

Question:Let A be the n x n matrix with 0's on the main diagonal, and 1's everywhere else. For an arbitrary vector b→in □n, solve the linear system Ax→=b⇀, expressing the components x1,.......,xnof x→in terms of the components of b⇀. See Exercise 69 for the case n=3 .

Short Answer

Expert verified

The solution of the linear system Ax→=b⇀ isxi=b1+......+bnn-1-bi, i=1,2,....,n .

Step by step solution

01

Consider the system.

IfA is an n x m matrix with row vectorsω1⇀,..........ωn⇀andx⇀is a vector in Rm then, .

Ax⇀=[-ω1⇀-...-ωn⇀-]x⇀=[-ω1⇀.x⇀-...-ωn⇀.x⇀-]

Consider the linear system.

y+z=ay+z=by+y=c

The matrix form of the system is,

011|1101|b110|c

The solution is, x=b+c-a2,y=a+c-b2,z=a+b-c2.

02

Compute the system

Consider the linear system, x1,.......,xnof x⇀x⇀in terms of the components ofb⇀.

x1+x2+.......+xn=b1x1+x2+.......+xn=b1.;x1+x2+.......+xn=bn-1x1+x2+.......+xn=bn

The solution, xi=b1+......+bnn-1-bi.

Where, i=1,2,....,n

Hence, xi=b1+......+bnn-1-bi, i=1,2,....,n is the solution of the linear systemAx→=b→.

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Most popular questions from this chapter

Explain why a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\) in \({\mathbb{R}^5}\) must be linearly independent when \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\) is linearly independent and \({{\mathop{\rm v}\nolimits} _4}\) is not in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

Find the polynomial of degree 2[a polynomial of the form f(t)=a+bt+ct2] whose graph goes through the points localid="1659342678677" (1,-1),(2,3)and(3,13).Sketch the graph of the polynomial.

Consider a dynamical system x→(t+1)=Ax→(t) with two components. The accompanying sketch shows the initial state vector x→0and two eigen vectors υ1→  and  υ2→ of A (with eigen values λ1→andλ2→ respectively). For the given values of λ1→andλ2→, draw a rough trajectory. Consider the future and the past of the system.

λ1→=1,λ2→=0.9

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)


Consider two vectors v→1 andv→2in R3 that are not parallel.

Which vectors inlocalid="1668167992227" ℝ3are linear combinations ofv→1andv→2? Describe the set of these vectors geometrically. Include a sketch in your answer.
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