This tool helps you determine if a vector is in the span of a set of vectors.
How to Use the Linear Algebra Span Calculator
To use the calculator, enter the vectors as a semicolon-separated list of comma-separated numbers. For example, to enter two vectors (1, 2, 3) and (4, 5, 6), type: 1,2,3;4,5,6.
Enter the point you want to check as a comma-separated list of numbers. For example, to check the point (1, 2, 3), type: 1,2,3.
Click the “Calculate” button to find out if the point is in the span of the vectors.
How It Works
The calculator uses Gaussian Elimination to row reduce the augmented matrix formed by adding the point as an extra column to the matrix of vectors. If the row-reduced form indicates that the point can be formed by a linear combination of the vectors, it outputs that the point is in the span; otherwise, it is not.
Limitations
This calculator is limited to finite-dimensional vector spaces and may have numerical stability issues for large or poorly scaled values due to the nature of JavaScript number handling.
Use Cases for This Calculator
Calculating the Span of a Single Vector
Enter the coefficients of a vector in the calculator to determine its span in the vector space. The calculator will analyze if the vector is linearly independent or dependent, providing you with insight into its span.
Finding the Span of Multiple Vectors
Input multiple vectors represented as rows of a matrix to find their collective span in the vector space. The calculator will apply linear algebra techniques to determine the dimensions of the vector space spanned by these vectors.
Verifying Linear Independence
Test a set of vectors for linear independence by entering them into the calculator. You will receive a result indicating whether the vectors are linearly independent or if there exists a linear combination resulting in the zero vector.
Assessing Linear Dependence
Check if a group of vectors is linearly dependent by inputting them into the calculator. The tool will assess whether one or more vectors in the set can be expressed as a linear combination of the others.
Determining Basis Vectors
Explore the basis vectors of a given vector space by inputting the relevant vectors into the calculator. It will help you identify the minimal set of linearly independent vectors that span the space.
Calculating Null Space
Analyze the null space of a matrix by entering its values into the calculator. You can determine the set of all vectors that map to zero when multiplied by the matrix, providing insights into its rank and nullity.
Identifying Column Space
Input a matrix to find its column space and understand the vectors in the space spanned by its columns. The calculator will help you identify the linear combinations of the column vectors that fill the entire space.
Exploring Row Space
Analyze the row space of a matrix to understand the vectors spanned by its rows. By entering the matrix values, you can determine the linear combinations of row vectors that cover the entire row space.
Checking Orthogonality
Verify the orthogonality of vectors by inputting them into the calculator. You will receive feedback on whether the vectors are orthogonal to each other, providing insights into their geometric properties.
Understanding Dimensionality
Explore the dimensionality of vector spaces by inputting vectors or matrices into the calculator. You can determine the maximum number of linearly independent vectors in the space and understand its overall complexity.