Use this tool to find the span of given vectors in linear algebra.

### How to Use the Linear Algebra Span Calculator

1. Enter the dimension of the vector space in the first input (e.g., 2 for 2-dimensional).

2. Enter the vectors in the second input box, each vector on a new line and elements separated by commas (e.g., “1,2” for a vector in 2D space).

3. Click “Calculate” to find the span of the entered vectors.

### Explanation

The span of a set of vectors in a vector space is the set of all linear combinations of those vectors. The script will take the entered vectors and calculate the possible linear combinations to determine the span.

### Limitations

- This calculator assumes that the vectors are entered correctly and are linearly independent.
- Currently, the calculator does not handle cases where more than one set of spanning vectors is possible.

## Use Cases for This Calculator

### Calculating Dot Product

Enter the two vectors in the provided input fields and click the “Calculate” button to get the dot product. The dot product of two vectors is found by multiplying their corresponding elements and then summing up the results. This is useful in determining the angle between two vectors and many other applications in linear algebra.

### Performing Matrix Multiplication

Input the matrices you want to multiply together and hit the “Calculate” button to obtain the resultant matrix. Matrix multiplication is a fundamental operation in linear algebra used in various fields such as computer graphics and solving systems of linear equations.

### Finding Determinant of a Matrix

Simply enter the matrix for which you want to find the determinant, and the tool will provide you with the result. The determinant of a matrix carries crucial information about the matrix’s properties, such as invertibility and solutions of linear systems.

### Solving Systems of Linear Equations

Input the coefficients of the equations and the constants on the RHS to determine the solution. This functionality is particularly useful in solving real-world problems where multiple linear equations need to be analyzed simultaneously.

### Computing Eigenvalues and Eigenvectors

Provide the matrix for which you want to find the eigenvalues and eigenvectors. This computation aids in understanding the behavior of linear transformations and stability analysis in various applications.

### Performing Matrix Inversion

Input the matrix you wish to invert, and the tool will output the inverted matrix. Matrix inversion is pivotal in numerous computations, including solving systems of linear equations and finding the least-squares solutions.

### Calculating Cross Product

Enter the two vectors to compute the cross product using the calculator. The cross product is a vector operation that results in a vector perpendicular to the plane containing the two input vectors, offering insights into geometry, physics, and more.

### Normalizing Vectors

Input a vector to normalize it, ensuring its magnitude becomes 1 while retaining its direction. Normalizing vectors is crucial in various applications, such as machine learning algorithms and computer graphics.

### Performing Scalar Multiplication

Input a scalar value and a vector to multiply them together efficiently. Scalar multiplication involves multiplying each element of a vector by a scalar, a fundamental operation in vector spaces and linear transformations.

### Finding Angle Between Vectors

Input the two vectors to determine the angle between them accurately. Knowing the angle between vectors provides insights into their relationship, orthogonality, and applications like robotics and navigation systems.