Is value in the matrix notation
$begingroup$
Given an $mtimes n$ matrix $A=begin{bmatrix} a_{1,1}&...&a_{1,n} \ vdots&ddots&vdots \ a_{m,1}&...&a_{m,n} \ end{bmatrix}$, say that I wanted to describe a set generator $S(A)$ which only consists of every individual entry in A, using set builder notation. I want to write something like this...
$$S(A) = left{sinmathbb{R} : sin Aright}$$
But I know that '$sin A$' is incorrect bc '$in$' should only ever be used wrt sets; is there a common convention (perhaps another symbol) that I could use?
linear-algebra notation
asked Jan 15 at 23:48
LandonLandon
123111
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add a comment |
$begingroup$
Given an $mtimes n$ matrix $A=begin{bmatrix} a_{1,1}&...&a_{1,n} \ vdots&ddots&vdots \ a_{m,1}&...&a_{m,n} \ end{bmatrix}$, say that I wanted to describe a set generator $S(A)$ which only consists of every individual entry in A, using set builder notation. I want to write something like this...
$$S(A) = left{sinmathbb{R} : sin Aright}$$
But I know that '$sin A$' is incorrect bc '$in$' should only ever be used wrt sets; is there a common convention (perhaps another symbol) that I could use?
linear-algebra notation
asked Jan 15 at 23:48
LandonLandon
123111
$endgroup$
add a comment |
$begingroup$
Given an $mtimes n$ matrix $A=begin{bmatrix} a_{1,1}&...&a_{1,n} \ vdots&ddots&vdots \ a_{m,1}&...&a_{m,n} \ end{bmatrix}$, say that I wanted to describe a set generator $S(A)$ which only consists of every individual entry in A, using set builder notation. I want to write something like this...
$$S(A) = left{sinmathbb{R} : sin Aright}$$
But I know that '$sin A$' is incorrect bc '$in$' should only ever be used wrt sets; is there a common convention (perhaps another symbol) that I could use?
linear-algebra notation
asked Jan 15 at 23:48
LandonLandon
123111
$endgroup$
Given an $mtimes n$ matrix $A=begin{bmatrix} a_{1,1}&...&a_{1,n} \ vdots&ddots&vdots \ a_{m,1}&...&a_{m,n} \ end{bmatrix}$, say that I wanted to describe a set generator $S(A)$ which only consists of every individual entry in A, using set builder notation. I want to write something like this...
$$S(A) = left{sinmathbb{R} : sin Aright}$$
But I know that '$sin A$' is incorrect bc '$in$' should only ever be used wrt sets; is there a common convention (perhaps another symbol) that I could use?
linear-algebra notation
linear-algebra notation
asked Jan 15 at 23:48
LandonLandon
123111
asked Jan 15 at 23:48
LandonLandon
123111
asked Jan 15 at 23:48
LandonLandon
123111
asked Jan 15 at 23:48
LandonLandon
123111
asked Jan 15 at 23:48
LandonLandon
123111
123111
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
What is a matrix, exactly?
A good formal definition of a matrix is a function
$$A : [m] times [n] rightarrow mathbb{R}$$
where $[m] = {1,2,cdots,m}$ and $[n] = {1,2,cdots,n}$. The codomain could be any set, but let's use $mathbb{R}$ for sake of example.
Thus, when you see an entry $a_{i,j}$, you can essentially think of it as the output $A(i,j) in mathbb{R}$. In this case, you can easily see that the notation you're looking for is simply the range $ran(A)$of $A$:
$$ ran(A) = {A(i,j) | (i,j) in [m]times [n]} = {a_{i,j} | (i,j) in [m] times [n]}$$
answered Jan 16 at 0:53
MetricMetric
1,23659
$endgroup$
$begingroup$
I strongly recommend not using $operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $mathbb R^n to mathbb R^m$. You can use the other two notations on the last line, though.
$endgroup$
– Eike Schulte
Jan 16 at 7:35
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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active
oldest
votes
$begingroup$
What is a matrix, exactly?
A good formal definition of a matrix is a function
$$A : [m] times [n] rightarrow mathbb{R}$$
where $[m] = {1,2,cdots,m}$ and $[n] = {1,2,cdots,n}$. The codomain could be any set, but let's use $mathbb{R}$ for sake of example.
Thus, when you see an entry $a_{i,j}$, you can essentially think of it as the output $A(i,j) in mathbb{R}$. In this case, you can easily see that the notation you're looking for is simply the range $ran(A)$of $A$:
$$ ran(A) = {A(i,j) | (i,j) in [m]times [n]} = {a_{i,j} | (i,j) in [m] times [n]}$$
answered Jan 16 at 0:53
MetricMetric
1,23659
$endgroup$
$begingroup$
I strongly recommend not using $operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $mathbb R^n to mathbb R^m$. You can use the other two notations on the last line, though.
$endgroup$
– Eike Schulte
Jan 16 at 7:35
add a comment |
$begingroup$
What is a matrix, exactly?
A good formal definition of a matrix is a function
$$A : [m] times [n] rightarrow mathbb{R}$$
where $[m] = {1,2,cdots,m}$ and $[n] = {1,2,cdots,n}$. The codomain could be any set, but let's use $mathbb{R}$ for sake of example.
Thus, when you see an entry $a_{i,j}$, you can essentially think of it as the output $A(i,j) in mathbb{R}$. In this case, you can easily see that the notation you're looking for is simply the range $ran(A)$of $A$:
$$ ran(A) = {A(i,j) | (i,j) in [m]times [n]} = {a_{i,j} | (i,j) in [m] times [n]}$$
answered Jan 16 at 0:53
MetricMetric
1,23659
$endgroup$
$begingroup$
I strongly recommend not using $operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $mathbb R^n to mathbb R^m$. You can use the other two notations on the last line, though.
$endgroup$
– Eike Schulte
Jan 16 at 7:35
add a comment |
$begingroup$
What is a matrix, exactly?
A good formal definition of a matrix is a function
$$A : [m] times [n] rightarrow mathbb{R}$$
where $[m] = {1,2,cdots,m}$ and $[n] = {1,2,cdots,n}$. The codomain could be any set, but let's use $mathbb{R}$ for sake of example.
Thus, when you see an entry $a_{i,j}$, you can essentially think of it as the output $A(i,j) in mathbb{R}$. In this case, you can easily see that the notation you're looking for is simply the range $ran(A)$of $A$:
$$ ran(A) = {A(i,j) | (i,j) in [m]times [n]} = {a_{i,j} | (i,j) in [m] times [n]}$$
answered Jan 16 at 0:53
MetricMetric
1,23659
$endgroup$
What is a matrix, exactly?
A good formal definition of a matrix is a function
$$A : [m] times [n] rightarrow mathbb{R}$$
where $[m] = {1,2,cdots,m}$ and $[n] = {1,2,cdots,n}$. The codomain could be any set, but let's use $mathbb{R}$ for sake of example.
Thus, when you see an entry $a_{i,j}$, you can essentially think of it as the output $A(i,j) in mathbb{R}$. In this case, you can easily see that the notation you're looking for is simply the range $ran(A)$of $A$:
$$ ran(A) = {A(i,j) | (i,j) in [m]times [n]} = {a_{i,j} | (i,j) in [m] times [n]}$$
answered Jan 16 at 0:53
MetricMetric
1,23659
answered Jan 16 at 0:53
MetricMetric
1,23659
answered Jan 16 at 0:53
MetricMetric
1,23659
answered Jan 16 at 0:53
MetricMetric
1,23659
1,23659
$begingroup$
I strongly recommend not using $operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $mathbb R^n to mathbb R^m$. You can use the other two notations on the last line, though.
$endgroup$
– Eike Schulte
Jan 16 at 7:35
add a comment |
$begingroup$
I strongly recommend not using $operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $mathbb R^n to mathbb R^m$. You can use the other two notations on the last line, though.
$endgroup$
– Eike Schulte
Jan 16 at 7:35
$begingroup$
I strongly recommend not using $operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $mathbb R^n to mathbb R^m$. You can use the other two notations on the last line, though.
$endgroup$
– Eike Schulte
Jan 16 at 7:35
$begingroup$
I strongly recommend not using $operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $mathbb R^n to mathbb R^m$. You can use the other two notations on the last line, though.
$endgroup$
– Eike Schulte
Jan 16 at 7:35
add a comment |
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