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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

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?

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

12.

If Ais an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.

Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

30.\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&{ - 2}&6\\0&{ - 5}&9\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&1&{ - 3}\\0&{ - 5}&9\end{array}} \right]\)

Let \(A\) be a \(3 \times 3\) matrix with the property that the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^3}\) into \({\mathbb{R}^3}\). Explain why transformation must be one-to-one.
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