independent column vector

bhuvaneshnick

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Code:
The maximum number  of linear independent column vectors of matrix A is called rank of matrix
i have not head this line ever,What does mean independent column vector .i would be nice to understand if u say what also dependent column vectors Thank you
 
Code:
The maximum number  of linear independent column vectors of matrix A is called rank of matrix
i have not head this line ever,What does mean independent column vector .i would be nice to understand if u say what also dependent column vectors Thank you

Do some google-search for "rank of matrix"

Start investigation with:

http://en.wikipedia.org/wiki/Rank_(linear_algebra)

Come back and tell us what you had learnt and where you are stuck.
 
Do some google-search for "rank of matrix"

Start investigation with:

http://en.wikipedia.org/wiki/Rank_(linear_algebra)

Come back and tell us what you had learnt and where you are stuck.

Yes i seen the page.i can able to find the rank of matrix by converting it into echelon form and fixing the rank by looking non zero rows
what i dont understand is finding rank by finding independent column vectors. The below mentioned is extracted from wikipedia
Code:
 1   2  1
-2 -3   1
 3   5   0
[COLOR=#252525][FONT=sans-serif]has rank 2: the first two rows are linearly independent, so the rank is at least 2, but all three rows are linearly dependent (the first is equal to the sum of the second and third) so the rank must be less than 3.
[/FONT][/COLOR]
it has been said that first two rows are independent so rank is 2 .i dont know how to find whether it is independent row vectors or dependent row vector and independent column vector or dependent column vector
 
Yes i seen the page.i can able to find the rank of matrix by converting it into echelon form and fixing the rank by looking non zero rows
what i dont understand is finding rank by finding independent column vectors. The below mentioned is extracted from wikipedia
Code:
 1   2  1
-2 -3   1
 3   5   0
[COLOR=#252525][FONT=sans-serif]has rank 2: the first two rows are linearly independent, so the rank is at least 2, but all three rows are linearly dependent (the first is equal to the sum of the second and third) so the rank must be less than 3.
[/FONT][/COLOR]
it has been said that first two rows are independent so rank is 2 .i dont know how to find whether it is independent row vectors or dependent row vector and independent column vector or dependent column vector

First, the page did not say 'it has been said that first two rows are independent so rank is 2.' The page said at least 2. It was only after it was noted that the first row was the sum of the second and third that the rank must be less than three and thus the rank was two.

Generally speaking, 'things' are linearly independent if no one of the 'things' can be written as a linear combination of the rest of the 'things' or, to put it a different way, if 'things' T1, T2, ..., and TN are linearly independent and
a1 T1 + a2 T2 + ... + an TN = [0]
then a1=a2=...=an=0. Note that the [0] is the zero 'thing'

In this particular case, the 'things' are matrix rows and we have
(-1) * Row1 + 1 * Row2 + 1 * Row3 = [0 0 0]
For your question, the 'things' are matrix columns and the [0] would be the zero column vector.
 
First, the page did not say 'it has been said that first two rows are independent so rank is 2.' The page said at least 2. It was only after it was noted that the first row was the sum of the second and third that the rank must be less than three and thus the rank was two.

Generally speaking, 'things' are linearly independent if no one of the 'things' can be written as a linear combination of the rest of the 'things' or, to put it a different way, if 'things' T1, T2, ..., and TN are linearly independent and
a1 T1 + a2 T2 + ... + an TN = [0]
then a1=a2=...=an=0. Note that the [0] is the zero 'thing'

In this particular case, the 'things' are matrix rows and we have
(-1) * Row1 + 1 * Row2 + 1 * Row3 = [0 0 0]
For your question, the 'things' are matrix columns and the [0] would be the zero column vector.


again little confusion

what do you saying for the below image.its saying that two rows are dependent
Capture2.JPG
 
I'm not sure where you got that information but it is incorrect. The last sentence should read "Then the third equation becomes a=0, a and b must be zero so the two rows are independent."

If fact, if you look at your post before this one, you have a matrix with the same two first rows, [1 2 1] and [-2 -3 1], and the statement is "the first two rows are linearly independent, so the rank is at least 2. ..." So you have shown two rows where one source has them (incorrectly) as dependent and the other source has them (correctly) as independent.
 
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