Question

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.

Short answer

1. False
2. False
3. True
4. True
5. False

Step-by-step solution

Identify whether the statement is true or false

a.

In the alternative notation, column vectors are written as (–4, 3) using parentheses and commas.

Thus, the given statement (a) is false.

Identify whether the statement is true or false

b.

If the vector’s corresponding points in the plane were on the same line through the origin, the vectors would have been scalar multiples of each other. The points in the plane corresponding to \(\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 5}\\2\end{array}} \right]\) are not scalar multiples of each other.

Thus, the given statement (b) is false.

Identify whether the statement is true or false

c.

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\({c_1},{c_2},...,{c_p}\).

Vector \(\frac{1}{2}{v_1} = \frac{1}{2}{v_1} + 0{v_2}\) is a linear combination of vectors \({v_1}\) and \({v_2}\).

Thus, the given statement (c) is true.

Identify whether the statement is true or false

d.

It is known that 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 \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{...}&{{a_n}}&b\end{array}} \right]\).

Thus, the given statement (d) is true.

Identify whether the statement is true or false

e.

The statement is frequently correct; however, span\(\left\{ {u,v} \right\}\)is not a plane when\(v\)is a multiple of\({\mathop{\rm u}\nolimits} \)or when\({\mathop{\rm u}\nolimits} \)is the zero vector.

Thus, the given statement (e) is false.