91Ó°ÊÓ

Use scalar multiplication and vector addition to rewrite the matrix into a vector equation \(\begin{aligned}{c}{x_1}\left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\0\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}0\\1\\2\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}5\\{ - 6}\\8\end{array}} \right] &= \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\\6\end{array}} \right]\\\left[ {\begin{array}{*{20}{c}}{{x_1} + 5{x_2}}\\{ - 2{x_1} + {x_2} - 6{x_3}}\\{2{x_2} + 8{x_3}}\end{array}} \right] &= \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\\6\end{array}} \right]\end{aligned}\).

The vectors on the left and right sides are equal if and only if their corresponding entries are equal. Thus,\({x_1}\)and\({x_2}\)make the vector equation\({x_1}{a_1} + {x_2}{a_2} = b\)if and only if\({x_1}\)and\({x_2}\)satisfy the system.

Write the matrix into a vector equation.

\(\begin{aligned}{c}{x_1} + 5{x_2} &= 2\\ - 2{x_1} + {x_2} - 6{x_3} &= - 1\\2{x_2} + 8{x_3} &= 6\end{aligned}\)

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

The augmented matrix for the vector equations \({x_1} + 5{x_2} = 2, - 2{x_1} + {x_2} - 6{x_3} = - 1\) and \(2{x_2} + 8{x_3} = 6\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}1&0&5&2\\{ - 2}&1&{ - 6}&{ - 1}\\0&2&8&6\end{array}} \right]\)

Perform an elementary row operation to produce the first augmented matrix.

Replace row 2 by adding 2 times row 1 to row 2

\(\left[ {\begin{array}{*{20}{c}}1&0&5&2\\{ - 0}&1&4&3\\0&2&8&6\end{array}} \right]\)

Perform an elementary row operation to produce a second augmented matrix.

Replace row 3 by adding - 2 times row 2 to row 3.

\(\left[ {\begin{array}{*{20}{c}}1&0&5&2\\0&1&4&3\\0&0&0&0\end{array}} \right]\)

The vector\({\mathop{\rm y}\nolimits} \)defined by\(y = {c_1}{v_1} + .... + {c_p}{v_p}\)is called alinear combination of\({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\)with weights\({c_1},{c_2},...,{c_p}\).

To obtain the solution of the system of equations, you have to convert the augmented matrix into the system of equations.

Write the obtained matrix \(\left[ {\begin{array}{*{20}{c}}1&0&5&2\\0&1&4&3\\0&0&0&0\end{array}} \right]\)into the equation notation.

\(\begin{array}{c}{x_1} + 5{x_2} = 2\\{x_2} + 4{x_3} = 3\end{array}\)

The system of equations corresponding to the vector equation \({x_1}{{\mathop{\rm a}\nolimits} _1} + {x_2}{{\mathop{\rm a}\nolimits} _2} + {x_3}{{\mathop{\rm a}\nolimits} _3} = {\mathop{\rm b}\nolimits} \) is consistent and has a solution. Hence, \({\mathop{\rm b}\nolimits} \) is a linear combination of columns \({{\mathop{\rm a}\nolimits} _1},{{\mathop{\rm a}\nolimits} _2}\), and \({{\mathop{\rm a}\nolimits} _3}\).

Thus, \({\mathop{\rm b}\nolimits} \) is a linear combination of columns \({{\mathop{\rm a}\nolimits} _1},{{\mathop{\rm a}\nolimits} _2}\), and \({{\mathop{\rm a}\nolimits} _3}\).