Linearly Dependent Rows and Rank Graphical Understanding
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I was having trouble today understanding the correlation between linearly dependent rows and the rank of a matrix. Based on my teaching, we can determine rank by solving for reduced row-echoleon form and simply count all the non-zero rows. This means all the fully 0 rows were linearly dependent on another and cancelled out.
Now, this idea would make much more sense to me if it referred to the columns of the matrix because I always pictured linear transformations as the unit vectors moving to the column vectors' coordinates. If two columns were colinear their span would become a line and thus the matrix output would lose a dimension.
However, reduce row echoleon uses linearly dependent rows to determine rank instead of columns. This idea does not quite make sense to me and I was hoping someone could clarify it based on my understanding.
Thanks
matrices matrix-rank
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
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add a comment |
$begingroup$
I was having trouble today understanding the correlation between linearly dependent rows and the rank of a matrix. Based on my teaching, we can determine rank by solving for reduced row-echoleon form and simply count all the non-zero rows. This means all the fully 0 rows were linearly dependent on another and cancelled out.
Now, this idea would make much more sense to me if it referred to the columns of the matrix because I always pictured linear transformations as the unit vectors moving to the column vectors' coordinates. If two columns were colinear their span would become a line and thus the matrix output would lose a dimension.
However, reduce row echoleon uses linearly dependent rows to determine rank instead of columns. This idea does not quite make sense to me and I was hoping someone could clarify it based on my understanding.
Thanks
matrices matrix-rank
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
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Row rank equals column rank. Row rank is number of nonzero rows in reduced row-echelon form, column rank is number of columns with a leading 1 in reduced row echelon form, and these are two ways of describing the same number.
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– Gerry Myerson
Jan 25 at 3:05
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Did that comment help any?
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– Gerry Myerson
Jan 26 at 5:10
1
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Earth to Mathew, come in, please.
$endgroup$
– Gerry Myerson
Jan 27 at 10:26
3
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I'm voting to close this question as off-topic because OP has abandoned it.
$endgroup$
– Gerry Myerson
Jan 29 at 8:28
add a comment |
$begingroup$
I was having trouble today understanding the correlation between linearly dependent rows and the rank of a matrix. Based on my teaching, we can determine rank by solving for reduced row-echoleon form and simply count all the non-zero rows. This means all the fully 0 rows were linearly dependent on another and cancelled out.
Now, this idea would make much more sense to me if it referred to the columns of the matrix because I always pictured linear transformations as the unit vectors moving to the column vectors' coordinates. If two columns were colinear their span would become a line and thus the matrix output would lose a dimension.
However, reduce row echoleon uses linearly dependent rows to determine rank instead of columns. This idea does not quite make sense to me and I was hoping someone could clarify it based on my understanding.
Thanks
matrices matrix-rank
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
$endgroup$
I was having trouble today understanding the correlation between linearly dependent rows and the rank of a matrix. Based on my teaching, we can determine rank by solving for reduced row-echoleon form and simply count all the non-zero rows. This means all the fully 0 rows were linearly dependent on another and cancelled out.
Now, this idea would make much more sense to me if it referred to the columns of the matrix because I always pictured linear transformations as the unit vectors moving to the column vectors' coordinates. If two columns were colinear their span would become a line and thus the matrix output would lose a dimension.
However, reduce row echoleon uses linearly dependent rows to determine rank instead of columns. This idea does not quite make sense to me and I was hoping someone could clarify it based on my understanding.
Thanks
matrices matrix-rank
matrices matrix-rank
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
asked Jan 25 at 2:46
Mathew SchauMathew Schau
1
1
$begingroup$
Row rank equals column rank. Row rank is number of nonzero rows in reduced row-echelon form, column rank is number of columns with a leading 1 in reduced row echelon form, and these are two ways of describing the same number.
$endgroup$
– Gerry Myerson
Jan 25 at 3:05
$begingroup$
Did that comment help any?
$endgroup$
– Gerry Myerson
Jan 26 at 5:10
1
$begingroup$
Earth to Mathew, come in, please.
$endgroup$
– Gerry Myerson
Jan 27 at 10:26
3
$begingroup$
I'm voting to close this question as off-topic because OP has abandoned it.
$endgroup$
– Gerry Myerson
Jan 29 at 8:28
add a comment |
$begingroup$
Row rank equals column rank. Row rank is number of nonzero rows in reduced row-echelon form, column rank is number of columns with a leading 1 in reduced row echelon form, and these are two ways of describing the same number.
$endgroup$
– Gerry Myerson
Jan 25 at 3:05
$begingroup$
Did that comment help any?
$endgroup$
– Gerry Myerson
Jan 26 at 5:10
1
$begingroup$
Earth to Mathew, come in, please.
$endgroup$
– Gerry Myerson
Jan 27 at 10:26
3
$begingroup$
I'm voting to close this question as off-topic because OP has abandoned it.
$endgroup$
– Gerry Myerson
Jan 29 at 8:28
$begingroup$
Row rank equals column rank. Row rank is number of nonzero rows in reduced row-echelon form, column rank is number of columns with a leading 1 in reduced row echelon form, and these are two ways of describing the same number.
$endgroup$
– Gerry Myerson
Jan 25 at 3:05
$begingroup$
Row rank equals column rank. Row rank is number of nonzero rows in reduced row-echelon form, column rank is number of columns with a leading 1 in reduced row echelon form, and these are two ways of describing the same number.
$endgroup$
– Gerry Myerson
Jan 25 at 3:05
$begingroup$
Did that comment help any?
$endgroup$
– Gerry Myerson
Jan 26 at 5:10
$begingroup$
Did that comment help any?
$endgroup$
– Gerry Myerson
Jan 26 at 5:10
1
1
$begingroup$
Earth to Mathew, come in, please.
$endgroup$
– Gerry Myerson
Jan 27 at 10:26
$begingroup$
Earth to Mathew, come in, please.
$endgroup$
– Gerry Myerson
Jan 27 at 10:26
3
3
$begingroup$
I'm voting to close this question as off-topic because OP has abandoned it.
$endgroup$
– Gerry Myerson
Jan 29 at 8:28
$begingroup$
I'm voting to close this question as off-topic because OP has abandoned it.
$endgroup$
– Gerry Myerson
Jan 29 at 8:28
add a comment |
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$begingroup$
Row rank equals column rank. Row rank is number of nonzero rows in reduced row-echelon form, column rank is number of columns with a leading 1 in reduced row echelon form, and these are two ways of describing the same number.
$endgroup$
– Gerry Myerson
Jan 25 at 3:05
$begingroup$
Did that comment help any?
$endgroup$
– Gerry Myerson
Jan 26 at 5:10
1
$begingroup$
Earth to Mathew, come in, please.
$endgroup$
– Gerry Myerson
Jan 27 at 10:26
3
$begingroup$
I'm voting to close this question as off-topic because OP has abandoned it.
$endgroup$
– Gerry Myerson
Jan 29 at 8:28