Question

if I have an mxn matrix A that does NOT have to be linearly independent and an affine subspace that's defined by (A^T)x = b, can it be proven that there always exists some combination of the columns of A which is part of that subspace?

Answer 1

can you guys help me with this?

Answer 2

What do you have so far?

Answer 3

nothing

Answer 4

i assume it is true, but i want to prove it

Answer 5

what is A^T ?

Answer 6

the transpose of A

Answer 7

ok!

Answer 8

I can think of a few examples where this statement is true, thus i assume it must be true for all examples, but this wouldn't be the mathematics category if I would leave it with this assumption.

Answer 9

from your equation:
\[\Large {A^T}x = b\]
we can say that the column b is a linear combination of the column of A^T

Answer 10

columns*

Answer 11

namely,
the column b is a linear combination of the rows of A

Answer 12

yes

Answer 13

now we can say that the rank of A is equal to:
\[\Large rank\left( A \right) \leqslant \min \left\{ {m,n} \right\}\]

Answer 14

is there something wrong with my browser or do you intend to use all these symbols? It is a pretty hard to read.

Answer 15

for example if I have a 3 x 2 matrix, then all what I can say is that the rank of my matrix is less or equal 2

Answer 16

yes

Answer 17

in other words, exists some linearly independent rows of A, which correspond to the same number of ilinearly independent columns of A^T, such that they generate the I mage of A as linear map.

Answer 18

linearly*

Answer 19

and those rows of A are a basis of a linear subspace

Answer 20

\[\left[\begin{matrix}a11 & a12 & ... & a1n \\ a21 & ... & ... & ... \\ ... & ... & ... & ... \\ an1 & ... & ... & ann\end{matrix}\right] *\left(\begin{matrix}x1 \\ x2 \\ ... \\ xn\end{matrix}\right)\]