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August 26th, 2015, 06:22 PM   #1
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Linear Independence and Linear Dependence.

Find n.m vectors LI in $M_{nXm}\left ( \mathbb{R} \right )$.
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August 27th, 2015, 10:10 AM   #2
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Do you know what $\displaystyle M_{n x m}$ means?

Last edited by Country Boy; August 27th, 2015 at 10:13 AM.
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August 28th, 2015, 08:16 AM   #3
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Square matrix ??
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August 28th, 2015, 08:17 AM   #4
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A generic matrix ????
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August 28th, 2015, 08:19 AM   #5
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He wants you to display mn (m times n) elements of that space such that they are linearly independent (whatever that means?).
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August 29th, 2015, 06:01 AM   #6
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You have posted a number of problems in linear algebra that depend, simply, on basic definitions. Here you have a question about a basis for a linear space, assert that they must be "linearly independent" and say "whatever that means". If you attempting problems like these and do not know what "basis" and "linearly independent" mean, you need to talk to your teacher about this!
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August 29th, 2015, 10:08 AM   #7
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I have no teacher, independent study ...
My friend sorry for my ignorance on the subject.

Last edited by Luiz; August 29th, 2015 at 10:17 AM.
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August 29th, 2015, 01:54 PM   #8
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That's not a problem, I'm just saying that you need to concentrate, at first, on learning the precise definitions.
In mathematics (as well as the sciences) definitions are working definitions- you use the precise words of the definitions in proving theorems and doing problems. To find a "basis" for a vector space you start by looking at the definition of basis.

Now, a "basis" for a vector space is a set of vectors so that any vector can be written, in a unique way, as a "linear combination" of the vectors in that set. "linear combination", in turn, means a sum of scalars (numbers) times the vectors. To find a basis for a vector space of matrices, you want a set of matrices, $\displaystyle \{M_i\}$, say, so that, for any matrix, M, there exist numbers, $\displaystyle a_i$ such that $\displaystyle M= a_1M_1+ a_2M_2+ \cdot\cdot\cdot+ a_nM_n$.

Start looking a simple example. m= n= 1 is just the same as the numbers {a}= a{1} for any "1 by 1 matrix"
The single "matrix", 1, is a basis and this space has dimension 1.

A 2 by 3 matrix is of the form
$\displaystyle \begin{bmatrix}a & b & c \\ d & e & f\end{bmatrix}= a\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}+ b\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}+ c\begin{bmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}+ d\begin{bmatrix}0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}+ e\begin{bmatrix}0 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}+ f\begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
Does that give you any ideas?
Thanks from Luiz

Last edited by Country Boy; August 29th, 2015 at 02:08 PM.
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