# Linear Independence Calculator

## Determine whether vectors are linearly independent step by step

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

Related calculator: Matrix Rank Calculator

Our Linear Independence Calculator serves as a practical digital resource for swiftly determining the linear independence or dependence of a given set of vectors. As a cornerstone principle in linear algebra and the realm of vector spaces, understanding and identifying linear independence is key to many complicated mathematical tasks.

## How to Use the Linear Independence Calculator?

### Input

Begin by introducing the coordinates of your vectors into the appropriate fields.

### Calculation

Proceed by clicking on the "Calculate" button. The calculator will then perform the calculation.

### Result

Finally, the calculator will present the results, notifying you if your set of vectors is linearly independent or dependent.

## What Are Linear Dependence and Independence?

Linear Dependence and Independence are fundamental concepts in the field of linear algebra and vector spaces.

**Linear Dependence**: This concept is central to linear algebra, referring to a situation where one or more vectors within a set can be defined as a linear combination of the others. The formula for linear combination is as follows:

where $$$\mathbf{\vec{v_1}},\mathbf{\vec{v_2}},\ldots,\mathbf{\vec{v_n}}$$$ are vectors and $$$a_1, a_2,\ldots,a_n$$$ are scalars (not all zero).

For instance, consider three vectors $$$\mathbf{\vec{v_1}}=\left[\begin{array}{c}1\\0\end{array}\right]$$$, $$$\mathbf{\vec{v_2}}=\left[\begin{array}{c}2\\0\end{array}\right]$$$, $$$\mathbf{\vec{v_3}}=\left[\begin{array}{c}3\\0\end{array}\right]$$$. Here, $$$\mathbf{\vec{v_2}}$$$ can be expressed as $$$2\mathbf{\vec{v_1}}$$$, and $$$\mathbf{\vec{v_3}}$$$ can be expressed as $$$3\mathbf{\vec{v_1}}$$$ or $$$\mathbf{\vec{v_1}}+\mathbf{\vec{v_2}}$$$. Hence, these vectors are linearly dependent.

**Linear Independence**: On the flip side, a set of vectors is described as linearly independent if no vector in the set can be expressed as a linear combination of the others. This is the case if the equation above holds only for $$$a_1=a_2=\ldots=a_n=0$$$.

For example, take two vectors $$$\mathbf{\vec{v_1}}=\left[\begin{array}{c}1\\0\end{array}\right]$$$ and $$$\mathbf{\vec{v_2}}=\left[\begin{array}{c}0\\1\end{array}\right]$$$. Here, no amount of scaling and adding $$$\mathbf{\vec{v_1}}$$$ can produce $$$\mathbf{\vec{v_2}}$$$, and vice versa. Hence, these vectors are linearly independent.

## How Can We Tell If Vectors Are Linearly Independent?

To determine if a set of vectors is linearly independent, follow these steps:

- Consider a set of vectors, $$$\mathbf{\vec{v_1}},\mathbf{\vec{v_2}},\ldots,\mathbf{\vec{v_n}}$$$, where $$$n$$$ is the number of vectors in the set.
- Formulate the linear combination equation using the scalars $$$a_1, a_2,\ldots,a_n$$$: $$$a_1\mathbf{\vec{v_1}}+a_2\mathbf{\vec{v_2}} +\ldots+a_n\mathbf{\vec{v_n}} =0$$$.
Set up a system of equations by equating the coordinates of the vectors to zero. This system can be represented as:

$$\begin{cases} a_1v_{11} +a_2v_{12}+\ldots+a_nv_{1n}=0\\a_1v_{21} +a_2v_{22}+\ldots+a_nv_{2n}=0\\\ldots\\a_1v_{n1} +a_2v_{n2}+\ldots+a_nv_{nn}=0\end{cases},$$where $$$v_{ij}$$$ is the $$$i$$$-th coordinate of the vector $$$\mathbf{\vec{v_j}}$$$.

- Solve the system of equations using a method such as Gauss-Jordan elimination. If the only solution is $$$a_1=a_2=\dots=a_n=0$$$ (the trivial solution), then the vectors are linearly independent. However, if there exist non-zero solutions for the scalars $$$a_1,a_2,\dots,a_n$$$, then the vectors are linearly dependent.

By examining the solutions of the system of equations, you can determine whether a set of vectors is linearly independent or dependent.

## Why Choose Our Linear Independence Calculator?

### Efficiency

Our calculator provides a quick and efficient way to determine the linear independence or dependence of a set of vectors. Instead of manually performing calculations, our calculator automates the process, saving you valuable time and effort.

### Accuracy

Our calculator uses reliable mathematical algorithms to accurately determine linear independence. By leveraging established techniques such as basis calculation or matrix rank determination, you can trust the results provided by our calculator.

### Versatility

Our calculator is designed to handle vectors of various dimensions. Whether you're working with two-dimensional or higher-dimensional vectors, our calculator can accommodate your needs and provide accurate results.

### Educational Resource

Our calculator not only provides the results of linear independence evaluation but also helps users understand the concept better. By showcasing the calculation steps and providing clear explanations, it serves as an educational resource to enhance your understanding of linear algebra concepts.

### FAQ

#### What is linear independence?

Linear independence refers to a set of vectors in which no vector can be expressed as a linear combination of the others. It means that each vector in the set contributes unique information or direction to the overall space.

#### What is the difference between linear independence and linear dependence?

Linear independence refers to a set of vectors where no vector can be expressed as a linear combination of the others. Linear dependence, on the other hand, indicates that at least one vector in the set can be written as a linear combination of the remaining vectors.

#### Can I use the calculator for both two-dimensional and higher-dimensional vectors?

Yes, the Linear Independence Calculator is designed to handle vectors of any dimension. It can be used for both two-dimensional and higher-dimensional vectors.

#### How many vectors can I input into the calculator?

The calculator can handle different number of vectors.