Nevíte-li si rady s jakýmkoliv matematickým problémem, toto místo je pro vás jako dělané.
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Hi ↑ stuart clark:,
, give ,
↑ stuart clark:
Let us define matrices
Let us assume that . Using vector notation we have
If then the vectors , would be linearly dependent and which is not true.
Considering the fact that we have .
On the other hand we have
If then the vectors , would be linearly dependent and .
Next let us define a matrix such that
Similarly as in the previous cases we have
From the fact that we deduce that the vectors , , are linearly dependent which implies
Hi ↑ stuart clark:,
As regards possible complements for the given solution, there is 2 ways.
The first one is to take advantage your first course of linear algebra (normally made in first year of university); and specially call back it two theorems
The determinant of a matrix M, of type (n, n) is no zero iff his rank is n.
A,B matrices of type (m,n),(n,m) resp.
and simply to apply.
(It is my method).
Other method, (for the high of grammar schools) is the one used by ↑ Pavel: but which is applicable only to this particular situation.
The method, I described former, can be used also for matrices of general type , resp. with not only in this particular case.
A lot of courage for your next exercise with n = 100 and m = 200.
Let A be a 100x200 matrix, B be a 200x100 matrix such that det(AB)=4. Prove that det(BA)=0.
Let , , and . Let us denote
Then the following identity holds true for :
Among 200 100-dimensional vectors , there exist at most 100 vectors , that are linearly independent. If then all the vectors would be linearly dependent and . Hence .
There is a similar identity for the vectors with :
For the same reasons as in the previous case we can deduce that if then all the vectors would be linearly dependent and . Hence .
We also have for
Due to the fact that we infer that the vectors , , are linearly dependent.
If you wish to examine another case, e.g. and , I will be glad to show you.
And it not much costs you to generalize your results and to put you to the same level that I.
Then well continuation.
( I notice that in analysis you used by the very complicated theorems and that in algebra on the contrary you always want to take back, algebraic demonstrations for every result)