RRE(A) - Reduced Row Echelon Form of A

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Chittesh14
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Here we have a matrix X which consists of the columns of A. So, in the question given: the rows of AT (Transpose of A) are the columns of A.
So, we have the basis of C which consists of the first two transposed rows of RRE(A).

Now, from what I know: N(A) i.e. the null space of A directly gives the linear combinations of the columns of A (i.e. shows any linear dependency between them). Now, we need to find the linear combinations of columns of A using the rows of At. N(At) i.e. the null space of A gives the linear combinations of the columns of At (i.e. rows of A – not columns of A). So, this won't help finding the link for the columns of A?

So, I need to find the rows of At or the columns of A as a linear combination of the first two rows of RRE(At). I can do this by inspection, but I want to know .... in general: How can you find the linear dependence of rows of a matrix using its RRE(A) i.e. reduced row echelon form?
Like, you can find the linear dependence of columns of A using N(A) : Ax = 0, how do you do that for RRE(A)?

Is there a link, and what is the use of RRE(A) for checking for linear dependence (if possible) e.g. c3, c4, ..., cn in terms of the basis vectors e.g. c1, c2?
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Chittesh14
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Chittesh14
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Anyone? :O
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Ryanzmw
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(Original post by Chittesh14)
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Here we have a matrix X which consists of the columns of A. So, in the question given: the rows of AT (Transpose of A) are the columns of A.
So, we have the basis of C which consists of the first two transposed rows of RRE(A).

Now, from what I know: N(A) i.e. the null space of A directly gives the linear combinations of the columns of A (i.e. shows any linear dependency between them). Now, we need to find the linear combinations of columns of A using the rows of At. N(At) i.e. the null space of A gives the linear combinations of the columns of At (i.e. rows of A – not columns of A). So, this won't help finding the link for the columns of A?

So, I need to find the rows of At or the columns of A as a linear combination of the first two rows of RRE(At). I can do this by inspection, but I want to know .... in general: How can you find the linear dependence of rows of a matrix using its RRE(A) i.e. reduced row echelon form?
Like, you can find the linear dependence of columns of A using N(A) : Ax = 0, how do you do that for RRE(A)?

Is there a link, and what is the use of RRE(A) for checking for linear dependence (if possible) e.g. c3, c4, ..., cn in terms of the basis vectors e.g. c1, c2?
Your inital rows span the row space but they're not a basis because they're not in general linear independent, the fact that you row reduce to get 2 rows of 0's means that those rows can be written in terms of the other two, which is literally what you're doing when you row reduce. Then because you can't row reduce further at the end it means that those row vectors are linear independent and will still span the row space and so form a basis for it

Hopefully you see how it is analogous for the column space.
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Chittesh14
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(Original post by Ryanzmw)
Your inital rows span the row space but they're not a basis because they're not in general linear independent, the fact that you row reduce to get 2 rows of 0's means that those rows can be written in terms of the other two, which is literally what you're doing when you row reduce. Then because you can't row reduce further at the end it means that those row vectors are linear independent and will still span the row space and so form a basis for it

Hopefully you see how it is analogous for the column space.
No, I understand this perfectly, thank you for the explanation though. I am trying to write each of the rows of At as a linear combination of the first two rows (that form a basis for the row space). I'm saying, how do I do that?

Nevermind, I get it. I read the question wrong to a certain extent, I kept trying to write A out as a column matrix and linking it to the rows, but it was much easier seeing it when you compare the rows of At to the basis vectors as in RRE(A) form. Thank you
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Chittesh14
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By the way, had 1 more question if you don't mind.

The RRE(A) tells us directly the linear combinations of the columns of A. E.g. Every column in RRE(A) can be written as a linear combination of the leading columns (columns that contain leading 1s) in RRE(A) and this linear combination is the same for the columns of A i.e. if column 3 of RRE(A) = 1 * column 1 of RRE(A) + 2 * column 2 of RRE(A), then column 3 of A = 1 * column 1 of A + 2 * column 2 of A.
So, for the columns of A, the linear dependence between the columns is the same as it is for RRE(A).

But, now from RRE(A) if I have a zero row, I know that row is linearly dependent on the rows which contain leading ones e.g. if I have three rows as in this example in this thread, I know the third row is linearly dependent on the first and second row. But, I can't find the linear dependence relation directly right, like r3 = r1 + r2 or whatever e.g. I can't see it directly from the RRE(A) right?
I can only see how the rows of A are linear combinations of the leading rows of RRE(A), but not linear combinations of other rows in A?

Sorry, it's hard to explain what I am trying to say .
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Ryanzmw
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(Original post by Chittesh14)
By the way, had 1 more question if you don't mind.

The RRE(A) tells us directly the linear combinations of the columns of A. E.g. Every column in RRE(A) can be written as a linear combination of the leading columns (columns that contain leading 1s) in RRE(A) and this linear combination is the same for the columns of A i.e. if column 3 of RRE(A) = 1 * column 1 of RRE(A) + 2 * column 2 of RRE(A), then column 3 of A = 1 * column 1 of A + 2 * column 2 of A.
So, for the columns of A, the linear dependence between the columns is the same as it is for RRE(A).

But, now from RRE(A) if I have a zero row, I know that row is linearly dependent on the rows which contain leading ones e.g. if I have three rows as in this example in this thread, I know the third row is linearly dependent on the first and second row. But, I can't find the linear dependence relation directly right, like r3 = r1 + r2 or whatever e.g. I can't see it directly from the RRE(A) right?
I can only see how the rows of A are linear combinations of the leading rows of RRE(A), but not linear combinations of other rows in A?

Sorry, it's hard to explain what I am trying to say .
You'd have to keep track of the operations you do in row reducing to know exactly how they relate (unless it's obvious), if I understand what you're asking
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Chittesh14
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(Original post by Ryanzmw)
You'd have to keep track of the operations you do in row reducing to know exactly how they relate (unless it's obvious), if I understand what you're asking
Yes that's fine . You have to keep track of how you reduce R1 using elementary row operations involving R2, R3 - but cannot tell directly from looking at RRE(A) form like you can for the columns. Great! !!!!! Thank you
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