Book a Demo!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutPoliciesSign UpSign In
Download
23127 views
ubuntu2004
Tweet
Share
Share
1
<exercise checkit-seed="0001" checkit-slug="V2" checkit-title="Linear combinations">

2
  <statement>

3
    <p>Consider the following statement.</p>

4
    <ul>

5
      <li>

6
            The vector <m> \left[\begin{array}{c}

7
-6 \\

8
1 \\

9
7 \\

10
-2

11
\end{array}\right] </m>is not

12
                a linear combination of the vectors <m> \left[\begin{array}{c}

13
4 \\

14
-2 \\

15
1 \\

16
0

17
\end{array}\right] , \left[\begin{array}{c}

18
-5 \\

19
0 \\

20
4 \\

21
-3

22
\end{array}\right] , \left[\begin{array}{c}

23
-3 \\

24
1 \\

25
-4 \\

26
-1

27
\end{array}\right] , \text{ and } \left[\begin{array}{c}

28
-2 \\

29
-2 \\

30
2 \\

31
-2

32
\end{array}\right] </m>.

33
            </li>

34
    </ul>

35
    <ol>

36
      <li> Write an equivalent statement using a vector equation.</li>

37
      <li> Explain why your statement is true or false.</li>

38
    </ol>

39
  </statement>

40
  <answer>

41
    <me>\operatorname{RREF}  \left[\begin{array}{cccc|c}

42
4 &amp; -5 &amp; -3 &amp; -2 &amp; -6 \\

43
-2 &amp; 0 &amp; 1 &amp; -2 &amp; 1 \\

44
1 &amp; 4 &amp; -4 &amp; 2 &amp; 7 \\

45
0 &amp; -3 &amp; -1 &amp; -2 &amp; -2

46
\end{array}\right] = \left[\begin{array}{cccc|c}

47
1 &amp; 0 &amp; 0 &amp; 0 &amp; -1 \\

48
0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\

49
0 &amp; 0 &amp; 1 &amp; 0 &amp; -1 \\

50
0 &amp; 0 &amp; 0 &amp; 1 &amp; 0

51
\end{array}\right] </me>

52
    <ol>

53
      <li> 

54
                The vector equation <m> x_{1} \left[\begin{array}{c}

55
4 \\

56
-2 \\

57
1 \\

58
0

59
\end{array}\right] + x_{2} \left[\begin{array}{c}

60
-5 \\

61
0 \\

62
4 \\

63
-3

64
\end{array}\right] + x_{3} \left[\begin{array}{c}

65
-3 \\

66
1 \\

67
-4 \\

68
-1

69
\end{array}\right] + x_{4} \left[\begin{array}{c}

70
-2 \\

71
-2 \\

72
2 \\

73
-2

74
\end{array}\right] = \left[\begin{array}{c}

75
-6 \\

76
1 \\

77
7 \\

78
-2

79
\end{array}\right] </m>has no solutions.</li>

80
      <li>

81
        <p><m> \left[\begin{array}{c}

82
-6 \\

83
1 \\

84
7 \\

85
-2

86
\end{array}\right] </m> is 

87
                 a linear combination of the vectors <m> \left[\begin{array}{c}

88
4 \\

89
-2 \\

90
1 \\

91
0

92
\end{array}\right] , \left[\begin{array}{c}

93
-5 \\

94
0 \\

95
4 \\

96
-3

97
\end{array}\right] , \left[\begin{array}{c}

98
-3 \\

99
1 \\

100
-4 \\

101
-1

102
\end{array}\right] , \text{ and } \left[\begin{array}{c}

103
-2 \\

104
-2 \\

105
2 \\

106
-2

107
\end{array}\right] </m>.

108
            </p>

109
      </li>

110
    </ol>

111
  </answer>

112
</exercise>

113


114


115