In linear algebra, a set of vectors is said to be linearly independent if none of the vectors can be expressed as a linear combination of the others. In other words, the only way to get the zero vector as a linear combination of the vectors is by setting all the coefficients to zero. Conversely, a set of vectors is said to be linearly dependent if at least one of the vectors can be expressed as a linear combination of the others. In this case, there are non-zero coefficients that can be used to obtain the zero vector as a linear combination of the vectors. In this article, we will discuss how to determine if a given set of vectors is linearly independent or linearly dependent.

Linear Independence

Let’s start with the definition of linear independence. A set of vectors {v1, v2, …, vn} is said to be linearly independent if the only solution to the equation:

c1v1 + c2v2 + … + cnvn = 0

is c1 = c2 = … = cn = 0. In other words, the only way to obtain the zero vector as a linear combination of the vectors is by setting all the coefficients to zero.

We can check for linear independence by constructing the augmented matrix:

\begin{bmatrix}v1 & v2 & \cdots & vn\end{bmatrix}

and row reducing it to reduced row echelon form. If the reduced matrix has n pivot columns, then the set of vectors is linearly independent. If the reduced matrix has fewer than n pivot columns, then the set of vectors is linearly dependent.

For example, let’s consider the set of vectors {v1, v2, v3} = {(1, 2, 3), (4, 5, 6), (7, 8, 9)}. To determine if this set is linearly independent, we construct the augmented matrix:

\begin{bmatrix}1 & 4 & 7\2 & 5 & 8\3 & 6 & 9\end{bmatrix}

We can row reduce this matrix to obtain the reduced row echelon form:

\begin{bmatrix}1 & 0 & -1\0 & 1 & 2\0 & 0 & 0\end{bmatrix}

Since there are only two pivot columns, we can conclude that the set of vectors {v1, v2, v3} is linearly dependent.

Linear Dependence

If a set of vectors is not linearly independent, then it is linearly dependent. This means that at least one of the vectors in the set can be expressed as a linear combination of the others. In other words, there are non-zero coefficients c1, c2, …, cn such that:

c1v1 + c2v2 + … + cnvn = 0

To determine which vector can be expressed as a linear combination of the others, we can use the same method as before. We construct the augmented matrix:

\begin{bmatrix}v1 & v2 & \cdots & vn\end{bmatrix}

and row reduce it to reduced row echelon form. If there is a row of zeros in the reduced matrix, then we can express one of the vectors as a linear combination of the others. The coefficients can be read off directly from the reduced matrix.

For example, let’s consider the set of vectors {v1, v2, v3} = {(1, 2, 3), (4, 5, 6), (7, 8, 9)}. We have already determined that this set is linearly dependent. To find which vector can be expressed as a linear combination of the others, we look at the reduced row echelon form:

\begin{bmatrix}1 & 0 & -1\0 & 1 & 2\0 & 0 & 0\end{bmatrix}

Since there is a row of zeros in the reduced matrix, we can express v3 as a linear combination of v1 and v2. From the matrix, we can read off the coefficients as c1 = -1 and c2 = -2:

v3 = -v1 – 2v2

Basis and Dimension

A basis for a vector space is a set of linearly independent vectors that span the space. This means that every vector in the space can be expressed as a linear combination of the basis vectors. A basis is also the smallest set of vectors that can span the space.

If a set of vectors is linearly independent, then it is a basis for the subspace spanned by those vectors. If a set of vectors is linearly dependent, then we can remove some of the vectors to obtain a basis. The maximum number of linearly independent vectors in a vector space is called the dimension of the space.

For example, let’s consider the set of vectors {v1, v2, v3} = {(1, 2, 3), (4, 5, 6), (7, 8, 9)}. We have already determined that this set is linearly dependent, and we have found that v3 can be expressed as a linear combination of v1 and v2. Therefore, we can remove v3 from the set to obtain a basis:

{(1, 2, 3), (4, 5, 6)}

This is a basis for the subspace spanned by v1, v2, and v3. Since there are two vectors in the basis, the dimension of the subspace is 2.

Geometric Interpretation

In two-dimensional space, a set of linearly independent vectors can be thought of as two arrows pointing in different directions. Any point in the plane can be reached by moving a certain distance in the direction of one arrow and a certain distance in the direction of the other arrow. If the vectors are linearly dependent, then one vector is a multiple of the other, and the arrows point in the same direction.

In three-dimensional space, a set of linearly independent vectors can be thought of as three arrows pointing in different directions. Any point in three-dimensional space can be reached by moving a certain distance in the direction of each arrow. If the vectors are linearly dependent, then at least one of the arrows lies in the plane formed by the other two arrows.

Conclusion

In this article, we have discussed how to determine if a given set of vectors is linearly independent or linearly dependent. We have shown that linear independence can be checked by row reducing the augmented matrix of the vectors, and that linear dependence can be detected by the presence of a row of zeros in the reduced matrix. We have also discussed the concepts of basis and dimension, and the geometric interpretation of linear independence in two and three-dimensional space. Linear independence is a fundamental concept in linear algebra and is used in many applications, such as determining the solutions to systems of linear equations and finding the eigenvalues and eigenvectors of a matrix.