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Chapter 1: Q31E (page 1)

Exercises 31 and 32 should be solved without performing row operations. [Hint: Write \(Ax = 0\) as a vector equation.]

31. Given \(A = \left[ {\begin{array}{*{20}{c}}2&3&5\\{ - 5}&1&{ - 4}\\{ - 3}&{ - 1}&{ - 4}\\1&0&1\end{array}} \right]\) . Observe that the third column is the sum of the first two columns. Find a nontrivial solution of \(Ax = 0\).

Short Answer

Expert verified

\(Ax = 0\) is a matrix equation for \(x = \left( {1,1, - 1} \right)\).

Step by step solution

01

Write the matrix as an expression

Matrix\(A\)can be written as the expression\(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right]\).

The third column is the sum of the first two columns, which indicates that \({a_3} = {a_1} + {a_2}\).

02

Rewrite the given equation

Rewrite the equation\({a_3} = {a_1} + {a_2}\)as\({a_1} + {a_2} - {a_3} = 0\).

Thus, \(Ax = 0\) is a matrix equation for \(x = \left( {1,1, - 1} \right)\).

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Most popular questions from this chapter

Suppose an experiment leads to the following system of equations:

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{249}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.843\end{aligned}\) (3)

  1. Solve system (3), and then solve system (4), below, in which the data on the right have been rounded to two decimal places. In each case, find the exactsolution.

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{25}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.8{\bf{4}}\end{aligned}\) (4)

  1. The entries in (4) differ from those in (3) by less than .05%. Find the percentage error when using the solution of (4) as an approximation for the solution of (3).
  1. Use your matrix program to produce the condition number of the coefficient matrix in (3).

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

Suppose the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer.

27. \(\begin{array}{l}{x_1} + 3{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

Find the general solutions of the systems whose augmented matrices are given as

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

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

15. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\)
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