When can I write a solution set in vector form as a span?
$begingroup$
I have a few questions about writing a solution set to a system of equations as a span.
- My first question is just a check to make sure I am not missing something fundamental. If my solution, in vector form, is a free variable times some vector $vec u$ can I write the solution as span{$vec u$}? For example:
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=0end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span$begin{Bmatrix}9\-3\1\0end{Bmatrix}$
- What happens if one or more basic variables cannot be expressed in terms of free variable(s) or $0$. For example: would I be able to describe the following solution set as a span given that $x_4=1$?
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=1end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $begin{bmatrix}0\0\0\1end{bmatrix}$
$+$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span{???}
- Finally, Is it possible to write a solution set as a span for a non-homogeneous system of equations?
linear-algebra matrices
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
$endgroup$
add a comment |
$begingroup$
I have a few questions about writing a solution set to a system of equations as a span.
- My first question is just a check to make sure I am not missing something fundamental. If my solution, in vector form, is a free variable times some vector $vec u$ can I write the solution as span{$vec u$}? For example:
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=0end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span$begin{Bmatrix}9\-3\1\0end{Bmatrix}$
- What happens if one or more basic variables cannot be expressed in terms of free variable(s) or $0$. For example: would I be able to describe the following solution set as a span given that $x_4=1$?
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=1end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $begin{bmatrix}0\0\0\1end{bmatrix}$
$+$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span{???}
- Finally, Is it possible to write a solution set as a span for a non-homogeneous system of equations?
linear-algebra matrices
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
$endgroup$
add a comment |
$begingroup$
I have a few questions about writing a solution set to a system of equations as a span.
- My first question is just a check to make sure I am not missing something fundamental. If my solution, in vector form, is a free variable times some vector $vec u$ can I write the solution as span{$vec u$}? For example:
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=0end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span$begin{Bmatrix}9\-3\1\0end{Bmatrix}$
- What happens if one or more basic variables cannot be expressed in terms of free variable(s) or $0$. For example: would I be able to describe the following solution set as a span given that $x_4=1$?
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=1end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $begin{bmatrix}0\0\0\1end{bmatrix}$
$+$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span{???}
- Finally, Is it possible to write a solution set as a span for a non-homogeneous system of equations?
linear-algebra matrices
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
$endgroup$
I have a few questions about writing a solution set to a system of equations as a span.
- My first question is just a check to make sure I am not missing something fundamental. If my solution, in vector form, is a free variable times some vector $vec u$ can I write the solution as span{$vec u$}? For example:
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=0end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span$begin{Bmatrix}9\-3\1\0end{Bmatrix}$
- What happens if one or more basic variables cannot be expressed in terms of free variable(s) or $0$. For example: would I be able to describe the following solution set as a span given that $x_4=1$?
$begin{align}
& x_1=9x_3\
& x_2=-3x_3\
& x_3=x_3\
& x_4=1end{align}$
$Rightarrow$
$begin{bmatrix}x_1\x_2\x_3\x_4end{bmatrix}$ $=$ $begin{bmatrix}0\0\0\1end{bmatrix}$
$+$ $x_3begin{bmatrix}9\-3\1\0end{bmatrix}$ $Rightarrow$ span{???}
- Finally, Is it possible to write a solution set as a span for a non-homogeneous system of equations?
linear-algebra matrices
linear-algebra matrices
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
asked Dec 28 '18 at 21:28
Gaussian EliminationGaussian Elimination
204
204
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add a comment |
1 Answer
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What $text{span}(...)$ means is that the solution set is the subspace spanned by the vectors you put inside of the parentheses. Thus, in order for you to be able to write the solution set in the $text{span}(...)$ format, the solution set has to be a subspace. However, by definition, a subspace has to contain the $0$ vector, which is only in the solution set of homogeneous systems of equations. Therefore, you can only write the solution in the $text{span}(...)$ format for homogeneous systems: It doesn't work for non-homogeneous systems because the solution sets for non-homogeneous systems aren't subspaces.
Thus, in Part 1, your solution in terms of $text{span}$ is correct, but you can't do the same thing for Part 2.
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
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1 Answer
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1 Answer
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$begingroup$
What $text{span}(...)$ means is that the solution set is the subspace spanned by the vectors you put inside of the parentheses. Thus, in order for you to be able to write the solution set in the $text{span}(...)$ format, the solution set has to be a subspace. However, by definition, a subspace has to contain the $0$ vector, which is only in the solution set of homogeneous systems of equations. Therefore, you can only write the solution in the $text{span}(...)$ format for homogeneous systems: It doesn't work for non-homogeneous systems because the solution sets for non-homogeneous systems aren't subspaces.
Thus, in Part 1, your solution in terms of $text{span}$ is correct, but you can't do the same thing for Part 2.
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
$endgroup$
add a comment |
$begingroup$
What $text{span}(...)$ means is that the solution set is the subspace spanned by the vectors you put inside of the parentheses. Thus, in order for you to be able to write the solution set in the $text{span}(...)$ format, the solution set has to be a subspace. However, by definition, a subspace has to contain the $0$ vector, which is only in the solution set of homogeneous systems of equations. Therefore, you can only write the solution in the $text{span}(...)$ format for homogeneous systems: It doesn't work for non-homogeneous systems because the solution sets for non-homogeneous systems aren't subspaces.
Thus, in Part 1, your solution in terms of $text{span}$ is correct, but you can't do the same thing for Part 2.
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
$endgroup$
add a comment |
$begingroup$
What $text{span}(...)$ means is that the solution set is the subspace spanned by the vectors you put inside of the parentheses. Thus, in order for you to be able to write the solution set in the $text{span}(...)$ format, the solution set has to be a subspace. However, by definition, a subspace has to contain the $0$ vector, which is only in the solution set of homogeneous systems of equations. Therefore, you can only write the solution in the $text{span}(...)$ format for homogeneous systems: It doesn't work for non-homogeneous systems because the solution sets for non-homogeneous systems aren't subspaces.
Thus, in Part 1, your solution in terms of $text{span}$ is correct, but you can't do the same thing for Part 2.
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
$endgroup$
What $text{span}(...)$ means is that the solution set is the subspace spanned by the vectors you put inside of the parentheses. Thus, in order for you to be able to write the solution set in the $text{span}(...)$ format, the solution set has to be a subspace. However, by definition, a subspace has to contain the $0$ vector, which is only in the solution set of homogeneous systems of equations. Therefore, you can only write the solution in the $text{span}(...)$ format for homogeneous systems: It doesn't work for non-homogeneous systems because the solution sets for non-homogeneous systems aren't subspaces.
Thus, in Part 1, your solution in terms of $text{span}$ is correct, but you can't do the same thing for Part 2.
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
answered Dec 28 '18 at 21:33
Noble MushtakNoble Mushtak
15.4k1835
15.4k1835
add a comment |
add a comment |
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