# Linear Algebra Calculator

## Solve linear algebra problems step by step

The Linear Algebra Calculator is designed to help you handle linear algebra problems. With an intuitive interface, you can quickly solve problems, check your solutions, and deepen your understanding of linear algebra concepts.

## How to use the Linear Algebra Calculator?

### Select a Calculator

Browse through the extensive list of linear algebra tools and click on the one that fits your needs.

### Input

Based on the calculator you've selected, fill in the required fields with the data you have.

### Calculation

Click the "Calculate" button.

### Result

Once you've inputted the necessary data and initiated the calculation, the calculator will process the information. After a brief moment, the computed solution will appear on the screen.

## What Is Linear Algebra?

Linear algebra is a broad and important mathematical discipline that studies vectors, vector spaces, and linear transformations acting on these spaces, as well as matrices and everything related to them.

This discipline is very important in various fields because its concepts are widely used:

Vectors

A vector is an ordered list of values. It can represent various physical quantities, such as force or velocity.

A vector $$$\mathbf{\vec{v}}$$$ in a two-dimensional space can be represented as follows:

$$\mathbf{\vec{v}}=\left[\begin{array}{c}v_1\\v_2\end{array}\right]$$Matrices

A matrix is a rectangular array of values. It can represent linear transformations or systems of linear equations.

A 2x2 matrix $$$A$$$ can be written as follows:

$$A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$$Linear Transformations and Matrices

Every linear transformation can be associated with a matrix. When a vector is multiplied by this matrix, it results in a transformed vector.

If $$$A$$$ is a matrix that represents a linear transformation and $$$\mathbf{\vec{v}}$$$ is a vector, then

$$A\mathbf{\vec{v}}=\mathbf{\vec{w}},$$where $$$\mathbf{\vec{w}}$$$ is the transformed vector.

Determinants

The determinant of a matrix gives a scalar value and has numerous applications, such as determining the invertibility of a matrix.

The formula for a 2x2 matrix $$$A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$$$:

$$\operatorname{det}(A)=ad-bc$$Eigenvalues and Eigenvectors

For a given matrix, an eigenvector is a non-zero vector that remains in the same direction when multiplied by the matrix. The corresponding scalar factor is the eigenvalue.

For a matrix $$$A$$$ and the eigenvector $$$\mathbf{\vec{v}}$$$ with the corresponding eigenvalue $$$\lambda$$$, we have that

$$A\mathbf{\vec{v}}=\lambda\mathbf{\vec{v}}$$

## What Calculators Does eMathHelp Offer?

### Add Matrices

This tool allows users to add matrices. Given two matrices, $$$A$$$ and $$$B$$$, of the same size, their sum, $$$A+B$$$, is found by adding the corresponding elements of the matrices.

### Add Vectors

This calculator helps find the sum of two vectors. The sum of vectors $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$ is another vector whose components are the sum of the corresponding components of $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$.

### Subtract Matrices

This subtracts the elements of one matrix from another, provided they have the same dimensions.

### Multiply Matrices

Matrix multiplication is a binary operation that produces a matrix from two matrices. For matrices $$$A$$$ of size $$$m\times n$$$ and $$$B$$$ of size $$$n\times p$$$, the resulting matrix will be of size $$$m\times p$$$.

### Divide Matrices

Matrix division isn't as direct as scalar division. Instead, we multiply one matrix by the inverse of another. If $$$A$$$ and $$$B$$$ are matrices, $$$A$$$ divided by $$$B$$$ is equivalent to $$$A$$$ multiplied by the inverse of $$$B$$$.

### Matrix Scalar Multiplication

Multiply every matrix element by a scalar (single number). If $$$\lambda$$$ is a scalar and $$$A$$$ is a matrix, then each entry of $$$\lambda A$$$ is $$$\lambda$$$ times the corresponding entry of $$$A$$$.

### Matrix Exponential

Given a square matrix $$$A$$$, the matrix exponential $$$e^A=\sum_{k=0}^{\infty}\frac{A^k}{k!}$$$.

### Matrix Power

This involves raising a square matrix to an integer power. For a matrix $$$A$$$, $$$A^2$$$ represents the matrix multiplication of $$$A$$$ with itself.

### Matrix Inverse

For a square matrix $$$A$$$, its inverse A^{-1} is the matrix such that when it's multiplied by $$$A$$$, the result is the identity matrix.

### Matrix of Minors

For each element in the matrix, remove its row and column, calculate the determinant of the resultant submatrix, and that's the minor for that element.

### Matrix Trace

The trace of a matrix is the sum of its diagonal elements.

### Matrix Transpose

Reflect a matrix over its main diagonal by swapping its rows and columns. The result is denoted as $$$A^T$$$.

### Matrix Determinant

This scalar value is obtained from a square matrix and is important in linear algebra, especially for systems of linear equations.

### Diagonalize Matrix

Convert a matrix into a diagonal form, where all off-diagonal elements are zero, if possible.

### LU Decomposition

Decompose a matrix into a product of a lower triangular matrix $$$L$$$ and an upper triangular matrix $$$U$$$.

### QR Factorization

Factor a matrix into a product of an orthogonal matrix $$$Q$$$ and an upper triangular matrix $$$R$$$.

### Moore-Penrose Inverse (Pseudoinverse)

This provides a way to compute a generalized matrix inverse when the matrix is not invertible.

### Cramer's Rule

This rule provides an explicit formula for the solution of a system of linear equations using matrix determinants.

### Subtract Vectors

Compute the difference of two vectors by subtracting corresponding components.

### Dot (Inner) Product

Given two vectors, the dot product is the sum of the products of their corresponding elements.

### Cross Product

For three-dimensional vectors, this gives a vector perpendicular to the two input vectors.

### Vector Scalar Multiplication

This is similar to matrix scalar multiplication but applied to vectors. Multiply every component of a vector by a scalar.

### Vector Magnitude

This is the length or size of a vector, computed as the square root of the sum of its components squared.

### Unit Vector

A vector of magnitude $$$1$$$, usually denoted as $$$\mathbf{\vec{e}}$$$, represents the direction of a given vector.

### Vector Projection

Project one vector onto another, resulting in a vector that is a "shadow" of one vector onto the other.

### Scalar Projection

This is the length of the vector projection.

### Triple Product

This involves three vectors and results in a scalar or vector value.

### FAQ

#### What is the Linear Algebra Calculator?

The Linear Algebra Calculator is an online tool that provides a comprehensive set of calculators designed to help users with a variety of linear algebra topics, from summing vectors to finding the pseudoinverse of a matrix.

#### How accurate is the Linear Algebra Calculator?

The calculator is designed with accuracy in mind. However, as with all computational tools, always double-check your results.

#### Can the calculator handle complex matrices or vectors?

Although the calculator is focused on real-number matrices and vectors, some functions can support complex numbers. Always check the instructions or notes associated with each specific calculator function.

#### How do the vector operations account for space? Are vectors in 2D, 3D, or higher dimensions?

Most vector operations are designed to be flexible and can handle vectors in 2D, 3D, and higher dimensions, provided the vectors meet the criteria for the operation (e.g., having the same dimension for addition).