Linear Dependence Calculator

How to Use the Linear Dependence Calculator

Step 1: Determine the Number of Vectors

Begin by deciding the number of vectors you want to analyze. You can input between 2 to 5 vectors. Enter this value in the Number of Vectors field.

Step 2: Set the Dimension of Vectors

Next, determine the dimension of your vectors. You can choose either 2 or 3 dimensions. Enter this value in the Vector Dimension field.

Step 3: Enter the Components of Each Vector

For each vector, enter its components according to the selected dimension. For two-dimensional vectors, you will need the x and y components. For three-dimensional vectors, enter the x, y, and z components.

• For Vector 1, enter values in the fields labeled Vector 1 (x component) and Vector 1 (y component), and if applicable, Vector 1 (z component).
• Repeat similar entries for Vector 2: Vector 2 (x component), Vector 2 (y component), and Vector 2 (z component).
• Continue this process for all vectors up to the number specified in Step 1.

Step 4: Analyze the Results

Once all vector components are entered, the calculator will automatically generate the results. Here are the key outputs:

• Determinant: This value is computed using the formula abs((vector1x * vector2y) - (vector1y * vector2x)). It determines the volume spanned by the vectors in two dimensions and indicates if vectors are linearly dependent.
• Rank of Matrix: This indicates the number of linearly independent rows in the matrix formed by the vectors. A rank of 2 implies the vectors are independent, while a rank of 1 suggests dependence.
• Linear Dependency Result: This field will declare the vectors as either “Linearly Independent” or “Linearly Dependent” based on the determinant.

Conclusion

Using these steps, you can efficiently determine the linear dependence of multiple vectors, which is particularly useful in various applications of linear algebra. Make sure to input accurate vector components to get reliable results.