Linear Algebra Ch 1.7 — Linear Independence

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Linear Algebra Ch 1.7 — Linear Independence 0/50
1 Linear Independence · Level 1
Determine if the vectors are linearly independent. Justify each answer. \(\mathbf{v}_1 = \vec{5, 1, 0}\), \(\mathbf{v}_2 = \vec{7, 2, -1}\), \(\mathbf{v}_3 = \vec{-2, 0, 6}\)
2 Linear Independence · Level 1
Determine if the vectors are linearly independent. Justify each answer. \(\mathbf{v}_1 = \vec{0, 0, 2}\), \(\mathbf{v}_2 = \vec{0, 5, -8}\), \(\mathbf{v}_3 = \vec{-3, 4, 1}\)
3 Linear Independence · Level 1
Determine if the vectors are linearly independent. Justify each answer. \(\mathbf{v}_1 = \vec{1, -3}\), \(\mathbf{v}_2 = \vec{-3, 6}\)
4 Linear Independence · Level 1
Determine if the vectors are linearly independent. Justify each answer. \(\mathbf{v}_1 = \vec{-1, 4}\), \(\mathbf{v}_2 = \vec{-2, 8}\)
5 Linear Independence · Level 2
Determine if the columns of the matrix form a linearly independent set. Justify each answer. \(\begin{pmatrix} 0 & -8 & 5 \\ 3 & -7 & 4 \\ -1 & 5 & -4 \\ 1 & -3 & 2 \end{pmatrix}\)
6 Linear Independence · Level 2
Determine if the columns of the matrix form a linearly independent set. Justify each answer. \(\begin{pmatrix} -4 & -3 & 0 \\ 0 & -1 & 4 \\ 1 & 0 & 3 \\ 5 & 4 & 6 \end{pmatrix}\)
7 Linear Independence · Level 2
Determine if the columns of the matrix form a linearly independent set. Justify each answer. \(\begin{pmatrix} 1 & 4 & -3 & 0 \\ -2 & -7 & 5 & 1 \\ -4 & -5 & 7 & 5 \end{pmatrix}\)
8 Linear Independence · Level 2
Determine if the columns of the matrix form a linearly independent set. Justify each answer. \(\begin{pmatrix} 1 & -3 & 3 & -2 \\ -3 & 7 & -1 & 2 \\ 0 & 1 & -4 & 3 \end{pmatrix}\)
9 Linear Independence · Level 3
(a) For what value(s) of \(h\) is \(\mathbf{v}_3\) in \(\text{Span}{\mathbf{v}_1, \mathbf{v}_2}\), and (b) for what value(s) of \(h\) is the set \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}\) linearly dependent? \(\mathbf{v}_1 = \vec{1, -3, 2}\), \(\mathbf{v}_2 = \vec{-3, 10, -6}\), \(\mathbf{v}_3 = \vec{2, -7, h}\)
10 Linear Independence · Level 3
(a) For what value(s) of \(h\) is \(\mathbf{v}_3\) in \(\text{Span}{\mathbf{v}_1, \mathbf{v}_2}\), and (b) for what value(s) of \(h\) is the set \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}\) linearly dependent? \(\mathbf{v}_1 = \vec{1, -5, -3}\), \(\mathbf{v}_2 = \vec{-2, 10, 6}\), \(\mathbf{v}_3 = \vec{2, -10, h}\)
11 Linear Independence · Level 2
Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer. \(\vec{1, -1, 4}\), \(\vec{3, -5, 7}\), \(\vec{-1, 5, h}\)
12 Linear Independence · Level 2
Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer. \(\vec{2, -4, 1}\), \(\vec{-6, 7, -3}\), \(\vec{8, h, 4}\)
13 Linear Independence · Level 2
Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer. \(\vec{1, 5, -3}\), \(\vec{-2, -9, 6}\), \(\vec{3, h, -9}\)
14 Linear Independence · Level 2
Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer. \(\vec{1, -3, 4}\), \(\vec{-6, 8, 7}\), \(\vec{4, h, -2}\)
15 Linear Independence · Level 1
Determine by inspection whether the vectors are linearly independent. Justify each answer. \(\vec{5, 1}\), \(\vec{2, 8}\), \(\vec{1, 3}\), \(\vec{-1, 7}\)
16 Linear Independence · Level 1
Determine by inspection whether the vectors are linearly independent. Justify each answer. \(\vec{4, -2, 6}\), \(\vec{6, -3, 9}\)
17 Linear Independence · Level 1
Determine by inspection whether the vectors are linearly independent. Justify each answer. \(\vec{3, 0, -1}\), \(\vec{0, 0, 0}\), \(\vec{-6, 5, 4}\)
18 Linear Independence · Level 1
Determine by inspection whether the vectors are linearly independent. Justify each answer. \(\vec{4, -3, 2, 8}\), \(\vec{-1, 3, 5, 1}\)
19 Linear Independence · Level 1
Determine by inspection whether the vectors are linearly independent. Justify each answer. \(\vec{-8, 12, -4}\), \(\vec{2, -3, -1}\)
20 Linear Independence · Level 1
Determine by inspection whether the vectors are linearly independent. Justify each answer. \(\vec{1, -7, 4}\), \(\vec{-2, 5, 3}\), \(\vec{0, 0, 0}\)
21 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (a) The columns of a matrix \(A\) are linearly independent if the equation \(A \mathbf{x} = \mathbf{0}\) has the trivial solution.
22 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (b) If \(S\) is a linearly dependent set, then each vector is a linear combination of the other vectors in \(S\).
23 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (c) The columns of any \(4 \times 5\) matrix are linearly dependent.
24 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (d) If \(\mathbf{x}\) and \(\mathbf{y}\) are linearly independent, and if \(\mathbf{z}\) is in \(\text{Span}{\mathbf{x}, \mathbf{y}}\), then \({\mathbf{x}, \mathbf{y}, \mathbf{z}}\) is linearly dependent.
25 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (e) If \(\mathbf{x}\) and \(\mathbf{y}\) are linearly independent, and if \({\mathbf{x}, \mathbf{y}, \mathbf{z}}\) is linearly dependent, then \(\mathbf{z}\) is in \(\text{Span}{\mathbf{x}, \mathbf{y}}\).
26 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (f) If a set in \(RR^n\) is linearly dependent, then the set contains more vectors than there are entries in each vector.
27 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (a) Two vectors are linearly dependent if and only if they lie on a line through the origin.
28 Linear Independence · Level 2
Mark each statement True or False. Justify each answer. (b) If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
29 Linear Independence · Level 3
Describe all possible reduced echelon forms of each matrix. \(A\) is a \(3 \times 3\) matrix with linearly independent columns.
30 Linear Independence · Level 3
Describe all possible reduced echelon forms of each matrix. \(A\) is a \(2 \times 2\) matrix with linearly dependent columns.
31 Linear Independence · Level 3
Describe all possible reduced echelon forms of each matrix. \(A\) is a \(4 \times 2\) matrix, \(A = \begin{pmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{pmatrix}\), and \(\mathbf{a}_2\) is not a multiple of \(\mathbf{a}_1\).
32 Linear Independence · Level 3
Describe all possible reduced echelon forms of each matrix. \(A\) is a \(4 \times 3\) matrix, \(A = \begin{pmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{pmatrix}\), where \({\mathbf{a}_1, \mathbf{a}_2}\) is linearly independent and \(\mathbf{a}_3\) is not in \(\text{Span}{\mathbf{a}_1, \mathbf{a}_2}\).
33 Linear Independence · Level 2
How many pivot columns must a \(7 \times 5\) matrix have if its columns are linearly independent? Why?
34 Linear Independence · Level 2
How many pivot columns must a \(5 \times 7\) matrix have if its columns span \(RR^5\)? Why?
35 Linear Independence · Level 3
Construct \(3 \times 2\) matrices \(A\) and \(B\) such that \(A \mathbf{x} = \mathbf{0}\) has only the trivial solution and \(B \mathbf{x} = \mathbf{0}\) has a nontrivial solution.
36 Linear Independence · Level 2
(a) Fill in the blank in the following statement: "If \(A\) is an \(m \times n\) matrix, then the columns of \(A\) are linearly independent if and only if \(A\) has ______ pivot columns."
(b) Explain why the statement in (a) is true.

Enter your answer directly below each part above.

37 Linear Independence · Level 3
Explain why the columns of the matrix \(A\) are linearly dependent, and find a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\) without row operations. \(A = \begin{pmatrix} 2 & 3 & 5 \\ -5 & 1 & -4 \\ -3 & -1 & -4 \\ 1 & 0 & 1 \end{pmatrix}\) [Note: The third column of \(A\) is the sum of the first two columns.]
38 Linear Independence · Level 3
Explain why the columns of the matrix \(A\) are linearly dependent, and find a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\) without row operations. \(A = \begin{pmatrix} 5 & 1 & 8 \\ -9 & 5 & 6 \\ 6 & -5 & -9 \end{pmatrix}\) [Note: The first column plus 3 times the second column equals the third column.]
39 Linear Independence · Level 3
Each statement is either true (in general) or false (provide a counterexample). (a) If \(\mathbf{v}_1, \ldots, \mathbf{v}_4\) are in \(RR^4\) and \(\mathbf{v}_3 = 2 \mathbf{v}_1 + \mathbf{v}_2\), then \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4}\) is linearly dependent.
40 Linear Independence · Level 3
Each statement is either true (in general) or false (provide a counterexample). (b) If \(\mathbf{v}_1, \ldots, \mathbf{v}_4\) are in \(RR^4\) and \(\mathbf{v}_3 = \mathbf{0}\), then \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4}\) is linearly dependent.
41 Linear Independence · Level 3
Each statement is either true (in general) or false (provide a counterexample). (c) If \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are in \(RR^4\) and \(\mathbf{v}_2\) is not a scalar multiple of \(\mathbf{v}_1\), then \({\mathbf{v}_1, \mathbf{v}_2}\) is linearly independent.
42 Linear Independence · Level 3
Each statement is either true (in general) or false (provide a counterexample). (d) If \(\mathbf{v}_1, \ldots, \mathbf{v}_4\) are in \(RR^4\) and \(\mathbf{v}_3\) is not a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_4\), then \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4}\) is linearly independent.
43 Linear Independence · Level 3
Each statement is either true (in general) or false (provide a counterexample). (e) If \(\mathbf{v}_1, \ldots, \mathbf{v}_4\) are in \(RR^4\) and \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}\) is linearly dependent, then \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4}\) is also linearly dependent.
44 Linear Independence · Level 3
Each statement is either true (in general) or false (provide a counterexample). (f) If \(\mathbf{v}_1, \ldots, \mathbf{v}_4\) are linearly independent vectors in \(RR^4\), then \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}\) is also linearly independent.
45 Linear Independence · Level 4
Suppose \(A\) is an \(m \times n\) matrix with the property that for all \(\mathbf{b}\) in \(RR^m\), the equation \(A \mathbf{x} = \mathbf{b}\) has at most one solution. Use the definition of linear independence to explain why the columns of \(A\) must be linearly independent.
46 Linear Independence · Level 4
Suppose an \(m \times n\) matrix \(A\) has \(n\) pivot columns. Explain why for each \(\mathbf{b}\) in \(RR^m\), the equation \(A \mathbf{x} = \mathbf{b}\) has at most one solution. [Hint: Explain why \(A \mathbf{x} = \mathbf{b}\) cannot have infinitely many solutions.]
47 Linear Independence · Level 4
Use as many columns of \(A\) as possible to construct a matrix \(B\) with the property that the equation \(B \mathbf{x} = \mathbf{0}\) has only the trivial solution. Solve \(B \mathbf{x} = \mathbf{0}\) to verify your work. \(A = \begin{pmatrix} 8 & -3 & 0 & -7 & 2 \\ -9 & 4 & 5 & 11 & -7 \\ 6 & -2 & 2 & -4 & 4 \\ 5 & -1 & 7 & 0 & 10 \end{pmatrix}\)
48 Linear Independence · Level 4
Use as many columns of \(A\) as possible to construct a matrix \(B\) with the property that the equation \(B \mathbf{x} = \mathbf{0}\) has only the trivial solution. Solve \(B \mathbf{x} = \mathbf{0}\) to verify your work. \(A = \begin{pmatrix} 12 & 10 & -6 & -3 & 7 & 10 \\ -7 & -6 & 4 & 7 & -9 & 5 \\ 9 & 9 & -9 & -5 & 5 & -1 \\ -4 & -3 & 1 & 6 & -8 & 9 \\ 8 & 7 & -5 & -9 & 11 & -8 \end{pmatrix}\)
49 Linear Independence · Level 4
With \(A\) and \(B\) as in Exercise 47, select a column \(\mathbf{v}\) of \(A\) that was not used in \(B\) and determine if \(\mathbf{v}\) is in the set spanned by the columns of \(B\).
50 Linear Independence · Level 4
With \(A\) and \(B\) as in Exercise 48, select a column \(\mathbf{v}\) of \(A\) that was not used in \(B\) and determine if \(\mathbf{v}\) is in the set spanned by the columns of \(B\).

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