Rank

•  rank

 

- An m x n matrix that consists of rows m and columns n.

- The rank of a matrix is the maximum number of linearly independent rows among m rows, or the maximum number of linearly independent columns among n columns.

- If every row is linearly independent, then the rank of the matrix is m.

- If all the columns are mutually independent, then the rank of the matrix is n.

- When every row and every column are linearly independent at the same time, the result is always the same, regardless of whether you look at it from the perspective of rows or columns.

- Knowing the rank of a matrix allows you to fully understand topics such as row-column independence, the dimension of the row space, theorems of dimensionality, invertibility, and basic row operations.

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• How to get the rank of this matrix :

 

\( A = \begin{pmatrix} 1 & 1 & 2 & -1 \\ 1 & -1 & 2 & -1 \end{pmatrix} \)

The column vectors are:

\( \mathbf{a}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \mathbf{a}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \quad \mathbf{a}_3 = 2\mathbf{a}_1, \quad \mathbf{a}_4 = -\mathbf{a}_1 \)

 

In this case, columns 1, 3, and 4 are all on the same straight line, so they are linearly dependent.
However, what about columns 1 and 2?

 

 

These two vectors are not scalar multiples of each other. Therefore, they are linearly independent.

 

We can summarize the situation as follows:

• Columns 1, 3, and 4: Since these vectors lie along the same line, we can select one of them.

• Column 2: Since this vector is linearly independent of the selected vector, we can select it as well.

Therefore, the maximum number of linearly independent column vectors is:

\[ \boxed{\operatorname{rank}(A)=2} \]

 

https://blog.naver.com/cindyvelyn/222139955990