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Chapter 1: Q21E (page 1)

Question 21: Let \({v_1} = \left( {\begin{array}{*{20}{c}}1\\0\\{ - 1}\\0\end{array}} \right),{v_2} = \left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\\1\end{array}} \right),{v_3} = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\) .

Short Answer

Expert verified

\(\left\{ {{v_1},{v_2},{v_3}} \right\}\) does not belong in span \({R^4}\).

Step by step solution

01

Reduce the matrix

First, reduce the row matrix\(\left\{ {{v_1},{v_2},{v_3}} \right\}\)to check whether it contains a pivot in each row.

So, the matrix \(\left\{ {{v_1},{v_2},{v_3}} \right\}\)can be written as:

\(\left\{ {{v_1},{v_2},{v_3}} \right\} = \left[ {\begin{array}{*{20}{c}}1&0&1\\0&{ - 1}&0\\{ - 1}&0&0\\0&1&{ - 1}\end{array}} \right]\)

02

Operations in rows

Apply \({R_3} \to {R_3} + {R_1}\) in the given matrix.

\(\left\{ {{v_1},{v_2},{v_3}} \right\} = \left[ {\begin{array}{*{20}{c}}1&0&1\\0&{ - 1}&0\\0&0&1\\0&1&{ - 1}\end{array}} \right]\)

Now, apply row operation \({R_4} \to {R_4} + {R_2}\)again.

\(\left\{ {{v_1},{v_2},{v_3}} \right\} = \left[ {\begin{array}{*{20}{c}}1&0&1\\0&{ - 1}&0\\0&0&1\\0&0&{ - 1}\end{array}} \right]\)

03

Resultant matrix

Now, apply row operation\({R_2} \to - {R_2}\)and \({R_4} \to {R_4} + {R_3}\).

\(\left\{ {{v_1},{v_2},{v_3}} \right\} = \left[ {\begin{array}{*{20}{c}}1&0&1\\0&1&0\\0&0&1\\0&0&0\end{array}} \right]\)

04

Result

From the above, it is concluded that the matrix \(\left\{ {{v_1},{v_2},{v_3}} \right\}\)does not contain a pivot in every row. As a result, span \({R^4}\) does not appear in the matrix column.

Hence, \(\left\{ {{v_1},{v_2},{v_3}} \right\}\)does not belong in span \({R^4}\).

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Most popular questions from this chapter

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

Question: Determine whether the statements that follow are true or false, and justify your answer.

16: There exists a 2x2 matrix such that

A[11]=[12]andA[22]=[21].

Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a plane in \({\mathbb{R}^3}\).

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

23.

a. Another notation for the vector \(\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\end{array}} \right]\) is \(\left[ {\begin{array}{*{20}{c}}{ - 4}&3\end{array}} \right]\).

b. The points in the plane corresponding to \(\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 5}\\2\end{array}} \right]\) lie on a line through the origin.

c. An example of a linear combination of vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) is the vector \(\frac{1}{2}{{\mathop{\rm v}\nolimits} _1}\).

d. The solution set of the linear system whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&b\end{array}} \right]\) is the same as the solution set of the equation\({{\mathop{\rm x}\nolimits} _1}{a_1} + {x_2}{a_2} + {x_3}{a_3} = b\).

e. The set Span \(\left\{ {u,v} \right\}\) is always visualized as a plane through the origin.

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)
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