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Chapter 1: Q15E (page 1)

Determine by inspection whether the vectors are linearly dependent. Justify each answer.

\(\left[ {\begin{array}{*{20}{c}}5\\1\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\8\end{array}} \right],\left[ {\begin{array}{*{20}{c}}1\\3\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 1}\\7\end{array}} \right]\)

Short Answer

Expert verified

The vectors are linearly dependent.

Step by step solution

01

Set of two or more vectors

When a set has more vectors than entries in each vector, it is said to be linearly dependent.

Let \({v_1},{v_2},\,{v_3}\), and \({v_4}\) be the four vectors. The linear dependence of these three vectors in the form of an augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}&{{v_3}}&{{v_4}}&0\end{array}} \right]\).

02

Augmented matrix

The augmented matrix is represented as:

\(\left[ {\begin{array}{*{20}{c}}5&2&1&{ - 1}&0\\1&8&3&7&0\end{array}} \right]\)

03

Linearly independent vector

There are four vectors given, but there are only two entries in each of the vectors. We know that if a set has more vectors than entries in each vector, it is linearly dependent.

Hence, the vectors are linearly dependent.

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Most popular questions from this chapter

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.

Suppose Tand U are linear transformations from \({\mathbb{R}^n}\) to \({\mathbb{R}^n}\) such that \(T\left( {U{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\) . Is it true that \(U\left( {T{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\)? Why or why not?

Consider the problem of determining whether the following system of equations is consistent:

\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase “columns of A.”
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of u and v. Is every vector in \({\mathbb{R}^2}\) a linear combination of u and v?

8.Vectors w, x, y, and z

In Exercises 9, write a vector equation that is equivalent to

the given system of equations.

9. \({x_2} + 5{x_3} = 0\)

\(\begin{array}{c}4{x_1} + 6{x_2} - {x_3} = 0\\ - {x_1} + 3{x_2} - 8{x_3} = 0\end{array}\)
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