What is the meaning of Rank[A | b]? (Linear Algebra)
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I got a question in my textbook where I am supposed to find if the linear system Ax=b is consistent. Then we are given some information. I do think I know how to solve this kind of problem but they use this notation Rank$[A | $b$]$ where it is some kind of number. What does this notation mean?
linear-algebra matrix-rank
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
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add a comment |
$begingroup$
I got a question in my textbook where I am supposed to find if the linear system Ax=b is consistent. Then we are given some information. I do think I know how to solve this kind of problem but they use this notation Rank$[A | $b$]$ where it is some kind of number. What does this notation mean?
linear-algebra matrix-rank
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
$endgroup$
add a comment |
$begingroup$
I got a question in my textbook where I am supposed to find if the linear system Ax=b is consistent. Then we are given some information. I do think I know how to solve this kind of problem but they use this notation Rank$[A | $b$]$ where it is some kind of number. What does this notation mean?
linear-algebra matrix-rank
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
$endgroup$
I got a question in my textbook where I am supposed to find if the linear system Ax=b is consistent. Then we are given some information. I do think I know how to solve this kind of problem but they use this notation Rank$[A | $b$]$ where it is some kind of number. What does this notation mean?
linear-algebra matrix-rank
linear-algebra matrix-rank
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
asked Dec 18 '18 at 14:11
J. DoeJ. Doe
627
627
add a comment |
add a comment |
2 Answers
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If the system of equations is written in matrix form, $Amathbf{x}=mathbf{b}$, then $A$ is the coefficient matrix and $[A|mathbf{b}]$ is the so called augmentend matrix: the matrix formed by adding a column to $A$, consisting of the constants $mathbf{b}$ from the system of equations (the right-hand side). You can check the Wikipedia page for some examples.
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
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add a comment |
$begingroup$
I'm guessing that $[A|b]$ refers to the augmented matrix formed by augmenting the column vector $b$ onto the matrix $A$. That is, it's a matrix with one extra column: $b$.
Note that the equation $Ax = b$ has a solution if and only if $b$ is in the columnspace of $A$ and $operatorname{Rank} [A | b] = operatorname{Rank} A$. Otherwise, $operatorname{Rank} [A | b] = operatorname{Rank} A + 1$.
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
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2 Answers
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2 Answers
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$begingroup$
If the system of equations is written in matrix form, $Amathbf{x}=mathbf{b}$, then $A$ is the coefficient matrix and $[A|mathbf{b}]$ is the so called augmentend matrix: the matrix formed by adding a column to $A$, consisting of the constants $mathbf{b}$ from the system of equations (the right-hand side). You can check the Wikipedia page for some examples.
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
$endgroup$
add a comment |
$begingroup$
If the system of equations is written in matrix form, $Amathbf{x}=mathbf{b}$, then $A$ is the coefficient matrix and $[A|mathbf{b}]$ is the so called augmentend matrix: the matrix formed by adding a column to $A$, consisting of the constants $mathbf{b}$ from the system of equations (the right-hand side). You can check the Wikipedia page for some examples.
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
$endgroup$
add a comment |
$begingroup$
If the system of equations is written in matrix form, $Amathbf{x}=mathbf{b}$, then $A$ is the coefficient matrix and $[A|mathbf{b}]$ is the so called augmentend matrix: the matrix formed by adding a column to $A$, consisting of the constants $mathbf{b}$ from the system of equations (the right-hand side). You can check the Wikipedia page for some examples.
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
$endgroup$
If the system of equations is written in matrix form, $Amathbf{x}=mathbf{b}$, then $A$ is the coefficient matrix and $[A|mathbf{b}]$ is the so called augmentend matrix: the matrix formed by adding a column to $A$, consisting of the constants $mathbf{b}$ from the system of equations (the right-hand side). You can check the Wikipedia page for some examples.
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
answered Dec 18 '18 at 14:15
StackTDStackTD
22.6k2049
22.6k2049
add a comment |
add a comment |
$begingroup$
I'm guessing that $[A|b]$ refers to the augmented matrix formed by augmenting the column vector $b$ onto the matrix $A$. That is, it's a matrix with one extra column: $b$.
Note that the equation $Ax = b$ has a solution if and only if $b$ is in the columnspace of $A$ and $operatorname{Rank} [A | b] = operatorname{Rank} A$. Otherwise, $operatorname{Rank} [A | b] = operatorname{Rank} A + 1$.
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
$endgroup$
add a comment |
$begingroup$
I'm guessing that $[A|b]$ refers to the augmented matrix formed by augmenting the column vector $b$ onto the matrix $A$. That is, it's a matrix with one extra column: $b$.
Note that the equation $Ax = b$ has a solution if and only if $b$ is in the columnspace of $A$ and $operatorname{Rank} [A | b] = operatorname{Rank} A$. Otherwise, $operatorname{Rank} [A | b] = operatorname{Rank} A + 1$.
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
$endgroup$
add a comment |
$begingroup$
I'm guessing that $[A|b]$ refers to the augmented matrix formed by augmenting the column vector $b$ onto the matrix $A$. That is, it's a matrix with one extra column: $b$.
Note that the equation $Ax = b$ has a solution if and only if $b$ is in the columnspace of $A$ and $operatorname{Rank} [A | b] = operatorname{Rank} A$. Otherwise, $operatorname{Rank} [A | b] = operatorname{Rank} A + 1$.
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
$endgroup$
I'm guessing that $[A|b]$ refers to the augmented matrix formed by augmenting the column vector $b$ onto the matrix $A$. That is, it's a matrix with one extra column: $b$.
Note that the equation $Ax = b$ has a solution if and only if $b$ is in the columnspace of $A$ and $operatorname{Rank} [A | b] = operatorname{Rank} A$. Otherwise, $operatorname{Rank} [A | b] = operatorname{Rank} A + 1$.
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
answered Dec 18 '18 at 14:15
Theo BenditTheo Bendit
17.3k12149
17.3k12149
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add a comment |
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