Linear Independence: Rows & Columns Of Matrices A & B

Let's dive into the concept of finding the maximum number of linearly independent rows and columns in matrices. This is a fundamental topic in linear algebra and has numerous applications in various fields such as engineering, computer science, and economics. Understanding linear independence is crucial for solving systems of equations, determining the rank of a matrix, and performing various matrix operations.

Understanding Linear Independence

Before we tackle the matrices provided, let's clarify what linear independence means. A set of vectors (in this case, rows or columns of a matrix) is said to be linearly independent if no vector in the set can be written as a linear combination of the others. In simpler terms, you can't create one vector by adding or scaling the other vectors in the set. If you can create one vector from the others, then the set is linearly dependent.

For example, consider two vectors: v1 = [1, 2] and v2 = [2, 4]. Here, v2 is simply 2 times v1. So, v1 and v2 are linearly dependent. However, if v2 = [2, 5], then v1 and v2 would be linearly independent because you can't get v2 by scaling v1.

Why is this important? Linear independence helps us understand the 'dimensionality' of the space spanned by the vectors. If the rows (or columns) of a matrix are linearly independent, they provide unique information and contribute to the matrix's rank. The rank of a matrix, which we'll touch on later, is the maximum number of linearly independent rows (or columns).

Keywords to keep in mind:

• Linear Independence: The property of a set of vectors where none can be expressed as a linear combination of the others.
• Linear Combination: A sum of scalar multiples of vectors.
• Rank of a Matrix: The maximum number of linearly independent rows or columns in a matrix.

Matrix A: Finding the Maximum Number of Linearly Independent Rows

Now, let's analyze the matrix A:

A =  \begin{pmatrix}
    -3 & -2 & 1 \\
    2 & 3 & -6 \\
    -1 & -4 & 5 \\
    1 & -1 & -1
\end{pmatrix}

To find the maximum number of linearly independent rows, we typically use row reduction techniques, such as Gaussian elimination or finding the row echelon form. The number of non-zero rows in the row echelon form tells us the rank of the matrix, which is the maximum number of linearly independent rows.

Steps to find the Rank:

1. Perform Row Operations: Use elementary row operations (swapping rows, multiplying a row by a scalar, adding a multiple of one row to another) to transform the matrix into row echelon form.
2. Identify Non-Zero Rows: Count the number of rows that have at least one non-zero element. These are the linearly independent rows.

Let's perform row operations on matrix A:

1. Swap Row 1 and Row 4: This puts a '1' in the top-left position, which is convenient for further row reduction.

   \begin{pmatrix}
       1 & -1 & -1 \\
       2 & 3 & -6 \\
       -1 & -4 & 5 \\
       -3 & -2 & 1
   \end{pmatrix}
   

2. Replace Row 2 with Row 2 - 2 * Row 1, Row 3 with Row 3 + Row 1, and Row 4 with Row 4 + 3 * Row 1:

   \begin{pmatrix}
       1 & -1 & -1 \\
       0 & 5 & -4 \\
       0 & -5 & 4 \\
       0 & -5 & -2
   \end{pmatrix}
   

3. Replace Row 3 with Row 3 + Row 2, and Row 4 with Row 4 + Row 2:

   \begin{pmatrix}
       1 & -1 & -1 \\
       0 & 5 & -4 \\
       0 & 0 & 0 \\
       0 & 0 & -6
   \end{pmatrix}
   

4. Swap Row 3 and Row 4:

   \begin{pmatrix}
       1 & -1 & -1 \\
       0 & 5 & -4 \\
       0 & 0 & -6 \\
       0 & 0 & 0
   \end{pmatrix}
   

We now have the row echelon form. There are 3 non-zero rows. Therefore, the maximum number of linearly independent rows in matrix A is 3. So, the rank of matrix A is 3.

Matrix B: Finding the Maximum Number of Linearly Independent Columns

Now, let's move on to matrix B:

B = \begin{pmatrix}
    9 & -3 & -1 \\
    -4 & 1 & 1
\end{pmatrix}

To find the maximum number of linearly independent columns, we can again use row reduction. However, this time, we're looking at the columns. We can transpose the matrix and perform row reduction, or we can perform column operations directly (though row operations are more common and often easier to implement).

Transpose Method:

1. Transpose Matrix B: This turns the columns into rows.

   B^T = \begin{pmatrix}
       9 & -4 \\
       -3 & 1 \\
       -1 & 1
   \end{pmatrix}
   

2. Perform Row Reduction: Find the row echelon form of B^T.

Direct Column Operations (Less Common):

Alternatively, we can perform column operations directly on matrix B. Elementary column operations include:

• Swapping columns
• Multiplying a column by a scalar
• Adding a multiple of one column to another

Let's use the transpose method and perform row reduction on B^T:

1. Divide Row 1 by 9:

   \begin{pmatrix}
       1 & -4/9 \\
       -3 & 1 \\
       -1 & 1
   \end{pmatrix}
   

2. Replace Row 2 with Row 2 + 3 * Row 1, and Row 3 with Row 3 + Row 1:

   \begin{pmatrix}
       1 & -4/9 \\
       0 & -1/3 \\
       0 & 5/9
   \end{pmatrix}
   

3. Multiply Row 2 by -3:

   \begin{pmatrix}
       1 & -4/9 \\
       0 & 1 \\
       0 & 5/9
   \end{pmatrix}
   

4. Replace Row 3 with Row 3 - (5/9) * Row 2:

   \begin{pmatrix}
       1 & -4/9 \\
       0 & 1 \\
       0 & 0
   \end{pmatrix}
   

We have 2 non-zero rows in the row echelon form of B^T. Therefore, the maximum number of linearly independent columns in matrix B is 2. So the rank of matrix B is 2.

Conclusion

In summary, the maximum number of linearly independent rows of matrix A is 3, and the maximum number of linearly independent columns of matrix B is 2. Understanding these concepts is crucial for various applications in linear algebra and related fields. Remember to use row reduction techniques (Gaussian elimination) to find the row echelon form of the matrix and count the number of non-zero rows to determine the rank and the maximum number of linearly independent rows or columns. Keep practicing, guys! You'll get the hang of it!