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Chapter 1: Q54E (page 37)


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.

Short Answer

Expert verified

The vectors of the form c1v→1+c2v→2a plane through the origin containing v→1and v→2.

Step by step solution

01

Step 1:

If vectors are not parallel then they are linearly independent. With the use parallelogram law of vector addition, we get that vector is linear combination of those 2 vectors if and only if it belongs to the same plane. Hence linear combination of the vectors v→1 and v→2 represents a plane i.e.

The vectors of the form c1v→1+c2v→2a plane through the origin containing v→1 and v→2.

02

Final answer:

The vectors of the form c1v→1+c2v→2 a plane through the origin containing v→1and v→2.

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Most popular questions from this chapter

Suppose Ais an \(n \times n\) matrix with the property that the equation \(A{\mathop{\rm x}\nolimits} = 0\) has at least one solution for each b in \({\mathbb{R}^n}\). Without using Theorem 5 or 8, explain why each equation Ax = b has in fact exactly one solution.

Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?

In Exercise 23 and 24, make each statement True or False. Justify each answer.

24.

a. Any list of five real numbers is a vector in \({\mathbb{R}^5}\).

b. The vector \({\mathop{\rm u}\nolimits} \) results when a vector \({\mathop{\rm u}\nolimits} - v\) is added to the vector \({\mathop{\rm v}\nolimits} \).

c. The weights \({{\mathop{\rm c}\nolimits} _1},...,{c_p}\) in a linear combination \({c_1}{v_1} + \cdot \cdot \cdot + {c_p}{v_p}\) cannot all be zero.

d. When are \({\mathop{\rm u}\nolimits} \) nonzero vectors, Span \(\left\{ {u,v} \right\}\) contains the line through \({\mathop{\rm u}\nolimits} \) and the origin.

e. Asking whether the linear system corresponding to an augmented matrix \(\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{\rm{1}}}}&{{{\rm{a}}_{\rm{2}}}}&{{{\rm{a}}_{\rm{3}}}}&{\rm{b}}\end{array}} \right]\) has a solution amounts to asking whether \({\mathop{\rm b}\nolimits} \) is in Span\(\left\{ {{a_1},{a_2},{a_3}} \right\}\).

Suppose \(a,b,c,\) and \(d\) are constants such that \(a\) is not zero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a,b,c,\) and \(d\)? Justify your answer.

28. \(\begin{array}{l}a{x_1} + b{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

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
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