Question

what is meaning of linear independent in matrices

Answer 1

Linearly independent really refers to vectors, not matrices. Vectors are linearly independent if there is no linear combination of the vectors that equals the zero vector, other than the trivial combination (zero of each vector). To bring that back to matrices, you would usually look at whether the column vectors or row vectors of a matrix are linearly independent.

Answer 2

thank you ,but may i get an example?

Answer 3

Of course! Here's an example of a matrix made of linearly independent vectors:
\[A=\left[\begin{matrix}
1&0&0\\0&1&0\\0&0&1
\end{matrix}\right]\]
You will of course recognize that as the identity matrix. The reason we know the vectors are linearly independent is because there is only one solution to Ax=0, and that solution is the zero vector. That is, for:
\[\left[\begin{matrix}
1&0&0\\0&1&0\\0&0&1
\end{matrix}\right]\cdot\left[\begin{matrix}
x_1\\x_2\\x_3
\end{matrix}\right]=\left[\begin{matrix}
0\\0\\0\\
\end{matrix}\right]
\]
the only solution is \(x_1=0, x_2=0, x_3=0\). You can try to find other values that would solve the equation, but you can't, because the vectors are linearly independent. For a counterexample, here's a matrix that has linearly dependent vectors:
\[A=\left[\begin{matrix}
1&2&3\\2&4&6\\3&6&9
\end{matrix}\right]\]
To see that it's linearly dependent, let's try that formula again, Ax=0.
\[\left[\begin{matrix}
1&2&3\\2&4&6\\3&6&9
\end{matrix}\right]\cdot\left[\begin{matrix}
x_1\\x_2\\x_3
\end{matrix}\right]=\left[\begin{matrix}
0\\0\\0\\
\end{matrix}\right]\\
\left[\begin{matrix}
1&2&3\\2&4&6\\3&6&9
\end{matrix}\right]\cdot\left[\begin{matrix}
5\\-1\\-1
\end{matrix}\right]=\left[\begin{matrix}
0\\0\\0\\
\end{matrix}\right]
\]
Since we were able to find non-zero solutions to the equation, the vectors are linearly dependent. An important intuition to gain here is that vectors are linearly dependent when they can be made out of the other vectors. In this example, the second column is just twice the first column, and the third column is just three times the first column. When one column is a multiple of another column, you know that the column vectors must be linearly dependent.

Answer 4

I should point out that the converse of that last statement is not true. Vectors do not have to be multiples of each other to be linearly dependent. For example, one vector could be the sum of two other vectors, and would be linearly dependent on those two vectors.