Linear Independence Calculator
Introduction
Our Linear Independence Calculator is a very handy and helpful online tool to find out whether a set of vector is linearly independent or dependent. It is based on the principle of linear algebra and the realm of vector spaces. It helps during solving many complicated mathematical tasks. Below you can see how to calculate linear independence with steps :
How to Use the Linear Independence Calculator?
1. select no of vectors and no. of coordinates from the drop-down in above tool.
2. Fill all the coordinates of all vectors.
3. Click on check independence and you will see your result that the set of vectors you entered is Linearly Independent or dependent.
4. You can change the coordinates or numbers of vectors or number of coordinates or you can click on clear to start from scratch again.
Understanding Vector Spaces and Linear Combinations
Vector spaces are collections of objects (vectors). We can addvector spaces together and scale it by numbers (scalars). They follow some specific rules, e.g. closure under addition and scalar multiplication.
A linear combination of vectors involves scaling each vector by a scalar and then adding them together. It's important to understand concepts like linear independence, span, and basis in vector spaces.
Span: Span is a set of all the possible combinations of a set of vectors.
Basis and Dimension: Basis is a set of independent vectors that span the entire space; dimension is the number of vectors in the basis.
Linear Independence: Vectors are independent if none can be written as a combination of the others.
Linear Dependence v/s Independence
Linear Dependence
This concept is based on linear algebra, it refers to a situation where one or more vectors within a set can be defined as a linear combination of the others. The formula for a linear combination is as follows:
\(a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \ldots + a_n \mathbf{v}_n = \mathbf{0},\)
where \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\) are vectors and \(a_1, a_2, \ldots, a_n\) are scalars (not all zero).
For example, consider three vectors \(\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\), \(\mathbf{v}_2 = \begin{bmatrix} 2 \\ 0 \end{bmatrix}\), \(\mathbf{v}_3 = \begin{bmatrix} 3 \\ 0 \end{bmatrix}\). Here, \(\mathbf{v}_2\) can be expressed as \(2\mathbf{v}_1\), and \(\mathbf{v}_3\) can be expressed as \(3\mathbf{v}_1\) or \(\mathbf{v}_1 + \mathbf{v}_2\). 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 \(\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\) and \(\mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\). Here, no amount of scaling and adding \(a_1 \mathbf{v}_1\) can produce \(\mathbf{v}_2\), and vice versa. Hence, these vectors are linearly independent.
How Can We Tell If Vectors Are Linearly Independent?
- Lets take a exmple of a set of vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \), where \( n \) is the number of vectors in the set.
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Lets put them in the formaula of linear combination equation using the scalars \( a_1,
a_2, \ldots, a_n \):
\[ a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \ldots + a_n \mathbf{v}_n = \mathbf{0} \] - Set up a system of equations by equating the coordinates of the vectors to zero. This system can be written as: \[ \begin{cases} a_1 v_{11} + a_2 v_{12} + \ldots + a_n v_{1n} = 0 \\ a_1 v_{21} + a_2 v_{22} + \ldots + a_n v_{2n} = 0 \\ \ldots \\ a_1 v_{n1} + a_2 v_{n2} + \ldots + a_n v_{nn} = 0 \end{cases} \]
- We can solve the system of equations using a method such as Gauss-Jordan elimination. If the only solution is \( a_1 = a_2 = \ldots = a_n = 0 \) (the trivial solution), then the vectors are linearly independent. However, if there exist non-zero solutions for the scalars \( a_1, a_2, \ldots, a_n \), then the vectors are linearly dependent.
By checking the solutions of the system of equations, you can determine whether a set of vectors is linearly independent or dependent.