91影视

In\({\mathbb{R}^2}\), the sum of two vectors\({\mathop{\rm u}\nolimits} \)and\({\mathop{\rm v}\nolimits} \)is thevector addition \({\mathop{\rm u}\nolimits} + v\), which is obtained by adding the corresponding entries of\({\mathop{\rm u}\nolimits} \)and\({\mathop{\rm v}\nolimits} \).

Thescalar multiple of a vector\({\mathop{\rm u}\nolimits} \)by real number\(c\)is the vector\(c{\mathop{\rm u}\nolimits} \)obtained by multiplying each entry in\({\mathop{\rm u}\nolimits} \)by\(c\).

Use scalar multiplicationto rewrite the matrix into a vector equation

\({x_1}\left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\2\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}0\\5\\5\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}2\\0\\8\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 5}\\{11}\\{ - 7}\end{array}} \right]\)

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 \({x_1}\left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\2\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}0\\5\\5\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}2\\0\\8\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 5}\\{11}\\{ - 7}\end{array}} \right]\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}1&0&2&{ - 5}\\{ - 2}&5&0&{11}\\2&5&8&{ - 7}\end{array}} \right]\)

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

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

\(\left[ {\begin{array}{*{20}{c}}1&0&2&{ - 5}\\0&5&4&1\\2&5&8&{ - 7}\end{array}} \right]\)

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

Sum of \( - 2\) times row 1 and row 3 at row 3

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

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

Sum of \( - 1\) times row 2 and row 3 at row 3

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

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 vector equations, you have to convert the augmented matrix into the system of equations.

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

\(\begin{array}{c}{x_1} + 2{x_3} = - 5\\5{x_2} + 4{x_3} = 1\\0 = 2\end{array}\)

The system of equations corresponding to the vector equation \({x_1}{{\mathop{\rm a}\nolimits} _1} + {x_2}{{\mathop{\rm a}\nolimits} _2} + {x_3}{{\mathop{\rm a}\nolimits} _3} = {\mathop{\rm b}\nolimits} \) has no solution. Hence, \({\mathop{\rm b}\nolimits} \) is not 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 not a linear combination of columns \({{\mathop{\rm a}\nolimits} _1},{{\mathop{\rm a}\nolimits} _2}\), and \({{\mathop{\rm a}\nolimits} _3}\).