# Calculators - Linear Algebra

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## Vector Addition Calculator

The calculator will find the sum of two vectors, with steps shown. It adds vectors of any size.

## Vector Subtraction Calculator

The calculator will find the difference of two vectors, with steps shown. It subtracts vectors of any size.

## Vector Scalar Multiplication Calculator

The calculator will multiply the given vector by the given scalar, with steps shown. It handles vectors of any size.

## Vector Dot (Inner) Product Calculator

An online calculator for finding the dot (inner) product of two vectors, with steps shown.

## Cross Product Calculator

The online calculator will find the cross product of two vectors, with steps shown.

## Vector Magnitude Calculator

An online calculator for finding the magnitude (length) of a vector, with steps shown.

## Unit Vector Calculator

The calculator will find the unit vector in the direction of the given vector, with steps shown.

## Angle between Vectors Calculator

The calculator will find the angle (in radians and degrees) between the two vectors and will show the work.

## Scalar Projection Calculator

The calculator will find the scalar projection of one vector onto another, with steps shown.

## Vector Projection Calculator

The calculator will find the vector projection of one vector onto another, with steps shown.

## Basis Calculator

The calculator will find a basis of the space spanned by the set of given vectors, with steps shown.

## Linear Independence Calculator

The calculator will determine whether the set of given vectors is linearly dependent or not, with steps shown.

## Gram-Schmidt Calculator

This calculator will orthonormalize the set of vectors, i.e. find the orthonormal basis, using the Gram-Schmidt process, with steps shown.

## Orthogonal Complement Calculator

This calculator will find the basis of the orthogonal complement of the subspace spanned by the given vectors, with steps shown.

## Triple Product Calculator

The calculator will calculate the triple product (both scalar and vector) of the three vectors, with steps shown.

## Vector Calculator

This calculator will add and subtract two vectors, with steps shown. It will also find the dot product, cross product, scalar projection, vector projection, angle betwen vectors, magnitude, unit vector, basis, orthonormal basis (using the Gram-Schmidt process), orthogonal complement, and triple product (both scalar and vector) and will multiply the vector by a scalar.

## Matrix Addition Calculator

The calculator will find the sum of two matrices (if possible), with steps shown. It adds matrices of any size up to 10x10 (2x2, 3x3, 4x4, etc.).

## Matrix Subtraction Calculator

The calculator will find the difference of two matrices (if possible), with steps shown. It subtracts matrices of any size up to 10x10 (2x2, 3x3, 4x4, etc.).

## Matrix Scalar Multiplication Calculator

The calculator will multiply the given matrix by the given scalar, with steps shown. It handles matrices of any size up to 10x10 (2x2, 3x3, 4x4, etc.).

## Matrix Multiplication Calculator

The calculator will find the product of two matrices (if possible), with steps shown. It multiplies matrices of any size up to 10x10 (2x2, 3x3, 4x4 etc.).

## Matrix Division Calculator

The calculator will find the quotient of two matrices (if possible), with steps shown. It divides matrices of any size up to 7x7 (2x2, 3x3, 4x4, etc.).

## Matrix Power Calculator

The calculator will find the given matrix raised to the given integer (positive or negative) power (if possible), with steps shown. Thus, it can square and cube the matrix. It handles matrices of any size up to 7x7 (2x2, 3x3, 4x4, etc.).

## Matrix Transpose Calculator

The calculator will find the transpose or the conjugate (Hermitian) transpose of the given matrix, with steps shown.

## Reduced Row Echelon Form (RREF) Calculator

The calculator will find the row echelon form (simple or reduced – RREF) of the given (augmented if needed) matrix, with steps shown.

## Gauss-Jordan Elimination Calculator

The calculator will perform the Gaussian elimination on the given augmented matrix, with steps shown. Complete reduction is available optionally.

## Column Space Calculator

The calculator will find the column space of the matrix, with steps shown.

## Matrix Null Space (Kernel) and Nullity Calculator

The calculator will find the null space (kernel) and the nullity of the given matrix, with steps shown.

## Matrix Determinant Calculator

The calculator will find the determinant of the matrix (2x2, 3x3, 4x4 etc.) using the cofactor expansion, with steps shown.

## Matrix Inverse Calculator

The calculator will find the inverse (if it exists) of the square matrix using the Gaussian elimination method or the adjugate method, with steps shown.

## Matrix of Minors Calculator

The calculator will find the matrix of minors of the given square matrix, with steps shown.

## Cofactor Matrix Calculator

The calculator will find the matrix of cofactors of the given square matrix, with steps shown.

## Adjoint Matrix Calculator

The calculator will find the adjoint (adjugate, adjunct) matrix of the given square matrix, with steps shown.

## Characteristic Polynomial Calculator

The calculator will find the characteristic polynomial of the given matrix, with steps shown.

## Eigenvalues and Eigenvectors Calculator

The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown.

## Diagonalize Matrix Calculator

The calculator will diagonalize the given matrix (if possible), with steps shown.

## Singular Value Decomposition Calculator

The calculator will find the singular value decomposition (SVD) of the given matrix, with steps shown.

## LU Decomposition Calculator

The calculator will find (if possible) the LU decomposition of the given matrix $$$A$$$, i.e. such a lower triangular matrix $$$L$$$ and an upper triangular matrix $$$U$$$ that $$$A=LU$$$, with steps shown.

In case of partial pivoting (permutation of rows is needed), the calculator will also find the permutation matrix $$$P$$$ such that $$$PA=LU$$$.

## QR Factorization Calculator

The calculator will find the QR factorization of the given matrix $$$A$$$, i.e. such an orthogonal (or semi-orthogonal) matrix $$$Q$$$ and an upper triangular matrix $$$R$$$ that $$$A=QR$$$, with steps shown.

## Matrix Calculator

This calculator will add, subtract, multiply, divide, and raise to power two matrices, with steps shown. It will also find the determinant, inverse, rref (reduced row echelon form), null space, rank, eigenvalues, and eigenvectors and will multiply the matrix by a scalar.