For the matrix: [[1,4,2],[2,5,1],[3,6,0]] (This is 3X3) .. find out whether the columns are linearly independent. Is the 3rd column a linear combination of two other columns? In both cases find the coefficients. Please show step-by-step and explain in great detail, so I can understand. Thank you.
They are dependent if the determinant is zero
Hi Steve, please explain if the determinant is zero. I am confused.
To determine whether the columns of a matrix are linearly independent and whether the third column is a linear combination of the other two columns, we can use the concept of matrix rank and matrix equations.
Step 1: Write the matrix in augmented form
Let's denote the matrix as A and its augmented form as [A|B], where B is a column vector of zeros since we are looking for a linear combination equal to the zero vector.
Matrix A:
```
[ 1 4 2 ]
[ 2 5 1 ]
[ 3 6 0 ]
```
Augmented form [A|B]:
```
[ 1 4 2 | 0 ]
[ 2 5 1 | 0 ]
[ 3 6 0 | 0 ]
```
Step 2: Perform row operations to obtain row-echelon form
We will perform elementary row operations (such as switching rows and multiplying rows by constants) to transform the augmented matrix into row-echelon form.
Row 2 - 2 * Row 1
Row 3 - 3 * Row 1
Updated matrix [A|B]:
```
[ 1 4 2 | 0 ]
[ 0 -3 -3 | 0 ]
[ 0 -6 -6 | 0 ]
```
Row 2 / -3
Row 3 - 2 * Row 2
Updated matrix [A|B]:
```
[ 1 4 2 | 0 ]
[ 0 1 1 | 0 ]
[ 0 0 0 | 0 ]
```
Step 3: Determine the rank of the matrix
The rank of a matrix is equal to the number of non-zero rows in its row-echelon form. In this case, the rank is 2 since there are 2 non-zero rows.
Step 4: Analyze the columns
If the rank of a matrix is less than the number of columns, it means that the columns are linearly dependent. Let's examine the rank of the original matrix and compare it to the number of columns.
In our case, the rank is 2, and the matrix has 3 columns. Therefore, the columns of the matrix are linearly dependent.
Step 5: Determine coefficients for linear combination
Since the columns are linearly dependent, we need to find the coefficients for the linear combination that results in the third column.
From the row-echelon form, we can observe the following equations:
1 * Column 1 + 4 * Column 2 = Column 3
This implies that the third column can be obtained by taking the first column and multiplying it by 1, and the second column and multiplying it by 4.
Therefore, the coefficients for the linear combination are 1 and 4, respectively.
To summarize:
- The columns of the matrix are linearly dependent.
- The third column is a linear combination of the other two columns with coefficients 1 and 4.
I hope this detailed explanation helps you understand the process of determining linear independence and finding coefficients for linear combinations in matrices. Let me know if you have any further questions!