Book a Demo!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutPoliciesSign UpSign In
Download
23127 views
ubuntu2004
Tweet
Share
Share
1


2
\begin{exercise}{V2}{Linear combinations}{0007}

3
\begin{exerciseStatement}

4


5
Consider the following statement.

6


7


8


9
\begin{itemize}

10
\item  The vector \( \left[\begin{array}{c}

11
8 \\

12
5 \\

13
-11 \\

14
11

15
\end{array}\right] \)is a linear combination of the vectors \( \left[\begin{array}{c}

16
0 \\

17
1 \\

18
3 \\

19
-1

20
\end{array}\right] , \left[\begin{array}{c}

21
3 \\

22
1 \\

23
-2 \\

24
1

25
\end{array}\right] , \left[\begin{array}{c}

26
-1 \\

27
-3 \\

28
-1 \\

29
-3

30
\end{array}\right] , \left[\begin{array}{c}

31
1 \\

32
-2 \\

33
-2 \\

34
0

35
\end{array}\right] , \text{ and } \left[\begin{array}{c}

36
4 \\

37
-1 \\

38
2 \\

39
1

40
\end{array}\right] \). 

41
\end{itemize}

42


43
    

44


45
\begin{enumerate}[(a)]

46
\item  Write an equivalent statement using a vector equation.

47
\item  Explain why your statement is true or false.

48
\end{enumerate}

49


50
    \end{exerciseStatement}

51
\begin{exerciseAnswer}\[\operatorname{RREF}  \left[\begin{array}{ccccc|c}

52
0 & 3 & -1 & 1 & 4 & 8 \\

53
1 & 1 & -3 & -2 & -1 & 5 \\

54
3 & -2 & -1 & -2 & 2 & -11 \\

55
-1 & 1 & -3 & 0 & 1 & 11

56
\end{array}\right] = \left[\begin{array}{ccccc|c}

57
1 & 0 & 0 & 0 & \frac{12}{5} & -3 \\

58
0 & 1 & 0 & 0 & -\frac{1}{5} & 2 \\

59
0 & 0 & 1 & 0 & -\frac{6}{5} & -2 \\

60
0 & 0 & 0 & 1 & \frac{17}{5} & 0

61
\end{array}\right] \]

62


63
\begin{enumerate}[(a)]

64
\item  The vector equation \( x_{1} \left[\begin{array}{c}

65
0 \\

66
1 \\

67
3 \\

68
-1

69
\end{array}\right] + x_{2} \left[\begin{array}{c}

70
3 \\

71
1 \\

72
-2 \\

73
1

74
\end{array}\right] + x_{3} \left[\begin{array}{c}

75
-1 \\

76
-3 \\

77
-1 \\

78
-3

79
\end{array}\right] + x_{4} \left[\begin{array}{c}

80
1 \\

81
-2 \\

82
-2 \\

83
0

84
\end{array}\right] + x_{5} \left[\begin{array}{c}

85
4 \\

86
-1 \\

87
2 \\

88
1

89
\end{array}\right] = \left[\begin{array}{c}

90
8 \\

91
5 \\

92
-11 \\

93
11

94
\end{array}\right] \)has a solution.

95
\item 

96


97
\( \left[\begin{array}{c}

98
8 \\

99
5 \\

100
-11 \\

101
11

102
\end{array}\right] \) is a linear combination of the vectors \( \left[\begin{array}{c}

103
0 \\

104
1 \\

105
3 \\

106
-1

107
\end{array}\right] , \left[\begin{array}{c}

108
3 \\

109
1 \\

110
-2 \\

111
1

112
\end{array}\right] , \left[\begin{array}{c}

113
-1 \\

114
-3 \\

115
-1 \\

116
-3

117
\end{array}\right] , \left[\begin{array}{c}

118
1 \\

119
-2 \\

120
-2 \\

121
0

122
\end{array}\right] , \text{ and } \left[\begin{array}{c}

123
4 \\

124
-1 \\

125
2 \\

126
1

127
\end{array}\right] \). 

128


129


130
\end{enumerate}

131


132
    \end{exerciseAnswer}

133
\end{exercise}

134


135


136