Linear Algebra Ch 1.5 — Solution Sets of Linear Systems

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Linear Algebra Ch 1.5 — Solution Sets of Linear Systems 0/52
1 Solution Sets of Linear Systems · Level 1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible. \(2 x_1 - 5 x_2 + 8 x_3 = 0\) \(-2 x_1 - 7 x_2 + x_3 = 0\) \(4 x_1 + 2 x_2 + 7 x_3 = 0\)
2 Solution Sets of Linear Systems · Level 1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible. \(x_1 - 3 x_2 + 7 x_3 = 0\) \(-2 x_1 + x_2 - 4 x_3 = 0\) \(x_1 + 2 x_2 + 9 x_3 = 0\)
3 Solution Sets of Linear Systems · Level 1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible. \(-3 x_1 + 5 x_2 - 7 x_3 = 0\) \(-6 x_1 + 7 x_2 + x_3 = 0\)
4 Solution Sets of Linear Systems · Level 1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible. \(-5 x_1 + 7 x_2 + 9 x_3 = 0\) \(x_1 - 2 x_2 + 6 x_3 = 0\)
5 Solution Sets of Linear Systems · Level 2
Solve the system and write the solution set in parametric vector form. \(x_1 + 3 x_2 + x_3 = 0\) \(-4 x_1 - 9 x_2 + 2 x_3 = 0\) \(-3 x_2 - 6 x_3 = 0\)
6 Solution Sets of Linear Systems · Level 2
Solve the system and write the solution set in parametric vector form. \(x_1 + 3 x_2 - 5 x_3 = 0\) \(x_1 + 4 x_2 - 8 x_3 = 0\) \(-3 x_1 - 7 x_2 + 9 x_3 = 0\)
7 Solution Sets of Linear Systems · Level 2
Describe all solutions of \(A \mathbf{x} = \mathbf{0}\) in parametric vector form, where \(A\) is row equivalent to the given matrix. \(\begin{pmatrix} 1 & 3 & -3 & 7 \\ 0 & 1 & -4 & 5 \end{pmatrix}\)
8 Solution Sets of Linear Systems · Level 2
Describe all solutions of \(A \mathbf{x} = \mathbf{0}\) in parametric vector form, where \(A\) is row equivalent to the given matrix. \(\begin{pmatrix} 1 & -2 & -9 & 5 \\ 0 & 1 & 2 & -6 \end{pmatrix}\)
9 Solution Sets of Linear Systems · Level 2
Describe all solutions of \(A \mathbf{x} = \mathbf{0}\) in parametric vector form, where \(A\) is row equivalent to the given matrix. \(\begin{pmatrix} 1 & -4 & -2 & 0 & 3 & -5 \\ 0 & 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
10 Solution Sets of Linear Systems · Level 2
Describe all solutions of \(A \mathbf{x} = \mathbf{0}\) in parametric vector form, where \(A\) is row equivalent to the given matrix. \(\begin{pmatrix} 1 & 5 & 2 & -6 & 9 & 0 \\ 0 & 0 & 1 & -7 & 4 & -8 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
11 Solution Sets of Linear Systems · Level 2
Describe all solutions of \(A \mathbf{x} = \mathbf{0}\) in parametric vector form, where \(A\) is row equivalent to the given matrix. \(\begin{pmatrix} 1 & -3 & -8 & 5 & 0 & -1 \\ 0 & 1 & 2 & -4 & 0 & 7 \\ 0 & 0 & 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
12 Solution Sets of Linear Systems · Level 2
Describe all solutions of \(A \mathbf{x} = \mathbf{0}\) in parametric vector form, where \(A\) is row equivalent to the given matrix. \(\begin{pmatrix} 1 & -2 & -8 & 3 & 0 & 6 \\ 0 & 0 & 1 & -5 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & -7 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
13 Solution Sets of Linear Systems · Level 2
Verify that the solutions found in Exercise 9 are indeed solutions of the corresponding homogeneous system. Multiply \(A\) times each solution vector to confirm \(A \mathbf{x} = \mathbf{0}\).
14 Solution Sets of Linear Systems · Level 2
Verify that the solutions found in Exercise 10 are indeed solutions of the corresponding homogeneous system. Multiply \(A\) times each solution vector to confirm \(A \mathbf{x} = \mathbf{0}\).
15 Solution Sets of Linear Systems · Level 2
Verify that the solutions found in Exercise 11 are indeed solutions of the corresponding homogeneous system. Multiply \(A\) times each solution vector to confirm \(A \mathbf{x} = \mathbf{0}\).
16 Solution Sets of Linear Systems · Level 2
Verify that the solutions found in Exercise 12 are indeed solutions of the corresponding homogeneous system. Multiply \(A\) times each solution vector to confirm \(A \mathbf{x} = \mathbf{0}\).
17 Solution Sets of Linear Systems · Level 2
The solution set of a certain linear system is described as follows: \(x_1 = 5 + 4 x_3\), \(x_2 = -2 - 7 x_3\), with \(x_3\) free. Use vectors to describe this solution set as a line in \(RR^3\).
18 Solution Sets of Linear Systems · Level 2
The solution set of a certain linear system is described as follows: \(x_1 = 3 x_4\), \(x_2 = 8 + x_4\), \(x_3 = 2 - 5 x_4\), with \(x_4\) free. Use vectors to describe this solution set as a line in \(RR^4\).
19 Solution Sets of Linear Systems · Level 3
Follow the method of Example 3 to describe the solutions of the following system in parametric vector form. Also, give a geometric description of the solution set and compare it to that in Exercise 5. \(x_1 + 3 x_2 + x_3 = 1\) \(-4 x_1 - 9 x_2 + 2 x_3 = -1\) \(-3 x_2 - 6 x_3 = -3\)
20 Solution Sets of Linear Systems · Level 3
Follow the method of Example 3 to describe the solutions of the following system in parametric vector form. Also, give a geometric description of the solution set and compare it to that in Exercise 6. \(x_1 + 3 x_2 - 5 x_3 = 4\) \(x_1 + 4 x_2 - 8 x_3 = 7\) \(-3 x_1 - 7 x_2 + 9 x_3 = -6\)
21 Solution Sets of Linear Systems · Level 2
Describe and compare the solution sets of \(x_1 + 9 x_2 - 4 x_3 = 0\) and \(x_1 + 9 x_2 - 4 x_3 = -2\).
22 Solution Sets of Linear Systems · Level 2
Describe and compare the solution sets of \(x_1 - 3 x_2 + 5 x_3 = 0\) and \(x_1 - 3 x_2 + 5 x_3 = 4\).
23 Solution Sets of Linear Systems · Level 2
Find the parametric equation of the line through \(\mathbf{a} = \vec{-2, 0}\) parallel to \(\mathbf{b} = \vec{-5, 3}\).
24 Solution Sets of Linear Systems · Level 2
Find the parametric equation of the line through \(\mathbf{a} = \vec{5, -2}\) parallel to \(\mathbf{b} = \vec{-4, 9}\).
25 Solution Sets of Linear Systems · Level 2
Find the parametric equation of the line \(M\) through \(\mathbf{p} = \vec{2, -5}\) and \(\mathbf{q} = \vec{-3, 1}\).
26 Solution Sets of Linear Systems · Level 2
Find the parametric equation of the line \(M\) through \(\mathbf{p} = \vec{-6, 3}\) and \(\mathbf{q} = \vec{0, -4}\).
27 Solution Sets of Linear Systems · Level 2
(T/F) A homogeneous equation is always consistent.
28 Solution Sets of Linear Systems · Level 2
(T/F) If x is a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\), then every entry in x is nonzero.
29 Solution Sets of Linear Systems · Level 2
(T/F) The equation \(A \mathbf{x} = \mathbf{0}\) gives an explicit description of its solution set.
30 Solution Sets of Linear Systems · Level 2
(T/F) The equation \(\mathbf{x} = x_2 \mathbf{u} + x_3 \mathbf{v}\), with \(x_2\) and \(x_3\) free (and neither \(\mathbf{u}\) nor \(\mathbf{v}\) a multiple of the other), describes a plane through the origin.
31 Solution Sets of Linear Systems · Level 2
(T/F) The homogeneous equation \(A \mathbf{x} = \mathbf{0}\) has the trivial solution if and only if the equation has at least one free variable.
32 Solution Sets of Linear Systems · Level 2
(T/F) The equation \(A \mathbf{x} = \mathbf{b}\) is homogeneous if the zero vector is a solution.
33 Solution Sets of Linear Systems · Level 2
(T/F) The equation \(\mathbf{x} = \mathbf{p} + t \mathbf{v}\) describes a line through \(\mathbf{v}\) parallel to \(\mathbf{p}\).
34 Solution Sets of Linear Systems · Level 2
(T/F) The effect of adding \(\mathbf{p}\) to a vector is to move the vector in a direction parallel to \(\mathbf{p}\).
35 Solution Sets of Linear Systems · Level 2
(T/F) The solution set of \(A \mathbf{x} = \mathbf{b}\) is the set of all vectors of the form \(\mathbf{w} = \mathbf{p} + \mathbf{v}_h\), where \(\mathbf{v}_h\) is any solution of \(A \mathbf{x} = \mathbf{0}\).
36 Solution Sets of Linear Systems · Level 2
(T/F) The solution set of \(A \mathbf{x} = \mathbf{b}\) is obtained by translating the solution set of \(A \mathbf{x} = \mathbf{0}\).
37 Solution Sets of Linear Systems · Level 3
Suppose that \(A \mathbf{x} = \mathbf{b}\) is consistent and let \(\mathbf{p}\) be a solution. Prove the second part of Theorem 6: Let \(\mathbf{w}\) be any solution of \(A \mathbf{x} = \mathbf{b}\), and define \(\mathbf{v}_h = \mathbf{w} - \mathbf{p}\). Show that \(\mathbf{v}_h\) is a solution of \(A \mathbf{x} = \mathbf{0}\), and hence that \(\mathbf{w} = \mathbf{p} + \mathbf{v}_h\).
38 Solution Sets of Linear Systems · Level 3
Suppose \(A \mathbf{x} = \mathbf{b}\) has a solution. Explain why the solution is unique precisely when \(A \mathbf{x} = \mathbf{0}\) has only the trivial solution.
39 Solution Sets of Linear Systems · Level 2
Suppose \(A\) is the \(3 \times 3\) zero matrix (all entries are zero). Describe the solution set of \(A \mathbf{x} = \mathbf{0}\).
40 Solution Sets of Linear Systems · Level 3
If \(\mathbf{b} \neq \mathbf{0}\), can the solution set of \(A \mathbf{x} = \mathbf{b}\) be a plane through the origin? Explain.
41 Solution Sets of Linear Systems · Level 3
Suppose \(A\) is a \(3 \times 3\) matrix with three pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have a solution for every \(\mathbf{b}\) in \(RR^3\)?
42 Solution Sets of Linear Systems · Level 3
Suppose \(A\) is a \(3 \times 3\) matrix with two pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have a solution for every \(\mathbf{b}\) in \(RR^3\)?
43 Solution Sets of Linear Systems · Level 3
Suppose \(A\) is a \(3 \times 4\) matrix with three pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have a solution for every \(\mathbf{b}\) in \(RR^3\)?
44 Solution Sets of Linear Systems · Level 3
Suppose \(A\) is a \(2 \times 4\) matrix with two pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have a solution for every \(\mathbf{b}\) in \(RR^2\)?
45 Solution Sets of Linear Systems · Level 2
Given \(A = \begin{pmatrix} -2 & -6 \\ 7 & 21 \\ -3 & -9 \end{pmatrix}\), find a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\) by inspection. [Hint: Think of the equation \(A \mathbf{x} = \mathbf{0}\) as a vector equation.]
46 Solution Sets of Linear Systems · Level 2
Given \(A = \begin{pmatrix} 4 & -6 \\ -8 & 12 \\ 6 & -9 \end{pmatrix}\), find a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\) by inspection. [Hint: Think of the equation \(A \mathbf{x} = \mathbf{0}\) as a vector equation.]
47 Solution Sets of Linear Systems · Level 3
Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\vec{1, 1, 1}\) is a solution of \(A \mathbf{x} = \mathbf{0}\).
48 Solution Sets of Linear Systems · Level 3
Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\vec{1, -2, 1}\) is a solution of \(A \mathbf{x} = \mathbf{0}\).
49 Solution Sets of Linear Systems · Level 4
Construct a \(2 \times 2\) matrix \(A\) such that the solution set of \(A \mathbf{x} = \mathbf{0}\) is the line in \(RR^2\) through \(\vec{4, 1}\) and the origin. Then, find a vector \(\mathbf{b}\) in \(RR^2\) such that the solution set of \(A \mathbf{x} = \mathbf{b}\) is NOT a line through \(\vec{4, 1}\) parallel to the solution set of \(A \mathbf{x} = \mathbf{0}\).
50 Solution Sets of Linear Systems · Level 4
Suppose \(A\) is a \(3 \times 3\) matrix and \(\mathbf{y}\) is a vector in \(RR^3\) such that the equation \(A \mathbf{x} = \mathbf{y}\) does not have a solution. Does there exist a vector \(\mathbf{z}\) in \(RR^3\) such that the equation \(A \mathbf{x} = \mathbf{z}\) has a unique solution? Discuss.
51 Solution Sets of Linear Systems · Level 3
Let \(A\) be an \(m \times n\) matrix and let \(\mathbf{u}\) be a vector in \(RR^n\) such that \(A \mathbf{u} = \mathbf{0}\). Show that for any scalar \(c\), the vector \(c \mathbf{u}\) is also a solution of \(A \mathbf{x} = \mathbf{0}\). [That is, show that \(A(c \mathbf{u}) = \mathbf{0}\).]
52 Solution Sets of Linear Systems · Level 3
Let \(A\) be an \(m \times n\) matrix and let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors in \(RR^n\) such that \(A \mathbf{u} = \mathbf{0}\) and \(A \mathbf{v} = \mathbf{0}\). Explain why \(A(\mathbf{u} + \mathbf{v}) = \mathbf{0}\). Then explain why \(A(c \mathbf{u} + d \mathbf{v}) = \mathbf{0}\) for any scalars \(c\) and \(d\).

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