# Linear Algebra Help

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#1
I understand that you can remove the third column in matrix A because the third column is column 1 and column 2 added together so there is nothing new.

However I do not understand how you could remove the first column.
How can you remove the first column? Isn't it not an addition of the other two columns?

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9 months ago
#2
(Original post by ChloeYeo)
I understand that you can remove the third column in matrix A because the third column is column 1 and column 2 added together so there is nothing new.

However I do not understand how you could remove the first column.
How can you remove the first column? Isn't it not an addition of the other two columns?

The picture that you've posted looks like it is showing the product of a 4x3 matrix with a 3x1 vector, which would give you a 4x1 vector. I think the lecturer is using b to refer to this entire 4x1 vector, so, for example, the first component of b would be x1 + x2 + 2x3. I'm not sure what you mean by "removing" columns from A - are you asking about something other than my interpretation?
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#3
Sorry I didn't make the question clear.

The lecturer asked "would removing column 1 give the same column space?" and said that the answer is "yes, it would".
I understand that removing column 3 would still give the same column space
however I don't understand how removing column 1 would give the same column space as well.

Thank you!
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9 months ago
#4
(Original post by ChloeYeo)
Sorry I didn't make the question clear.

The lecturer asked "would removing column 1 give the same column space?" and said that the answer is "yes, it would".
I understand that removing column 3 would still give the same column space
however I don't understand how removing column 1 would give the same column space as well.

Thank you!
Surely they mean one or the other
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#5
(Original post by Meowstic)
Surely they mean one or the other
Why??
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9 months ago
#6
(Original post by ChloeYeo)
Why??
It doesn't matter which you remove why would it have to be the third one
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#7
(Original post by Meowstic)
It doesn't matter which you remove why would it have to be the third one
Isn't because the third column is the addition of first and second column??
Isn't that the reason why removing the third column will still give us the same column space?
Thank you.
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9 months ago
#8
(Original post by ChloeYeo)
Isn't because the third column is the addition of first and second column??
Isn't that the reason why removing the third column will still give us the same column space?
Thank you.
The third column is a linear combination of the first and second, yes so you can remove it.
The second column is therefore necessarily a linear combination of the first and third, so you can remove it.
The first column is necessarily a linear combination of the second and third so you can remove it.
You only need two vectors because your space is 2d, doesn't matter what these vectors are as long as they span the space.
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#9
(Original post by Meowstic)
The third column is a linear combination of the first and second, yes so you can remove it.
The second column is therefore necessarily a linear combination of the first and third, so you can remove it.
The first column is necessarily a linear combination of the second and third so you can remove it.
You only need two vectors because your space is 2d, doesn't matter what these vectors are as long as they span the space.
OHH so is subtraction a linear combination as well as addition?

Why is my space 2d? I thought it was 4d because there were 4 rows.

Thank you!!
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9 months ago
#10
(Original post by ChloeYeo)
OHH so is subtraction a linear combination as well as addition?

Why is my space 2d? I thought it was 4d because there were 4 rows.

Thank you!!
Three vectors e1 e2 e3.
If e3 is a linear combination of e1 and e2 then there exists some values a and b such that
e3 = a e1 + b e2
Clearly it isn't necessary to use e3 as you can just replace it with e1 and e2 in that combination. It's also clear that this choice of vector to "remove" isn't unique for non zero a and b, you can just rearrange the above equation.

The subspace spanned by vectors e1 and e2 must be two dimensional as you only have two degrees of freedom (pretty much the definition of what it means to be 2d). It's just embedded in a four dimensional space. If you draw two vectors on a piece of paper then any combination of those two vectors is still on the paper, even if the paper itself is in our three dimensional world anything that lives on the surface of it, only moving along those two vectors, can only ever know two.

The matrix in the example you posted takes any vector x in a 3d space then maps it to a 4d space, however if you consider all inputs, you will find that the corresponding outputs span a 2d subspace of the 4d space that you can't ever get out of. This is due to the degeneracy in the columns.

If you ignore that, then it should still be obvious that the outputs of the matrix can only describe a 3d subspace at most as, if you consider the matrix to map a point in the input to a point in the output, it wouldn't make sense at all if you could map every point in a 4d space to a unique point in 3d space because then you would only need 3 coordinates to describe a point in 4d which is ridiculous.
Last edited by username3331778; 9 months ago
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9 months ago
#11
(Original post by ChloeYeo)
OHH so is subtraction a linear combination as well as addition?...
Look up the definition of "Linear Combination":

https://mathworld.wolfram.com/LinearCombination.html

note that a, b,c are constants.
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