Linear correlation is an important concept in linear algebra, used to describe the linear relationship between vectors or groups of vectors. The following are several ways to judge whether a vector group is linearly related:
1. **Judgment by definition**:
- If there is a set of coefficients that are not all zero, such that the sum of these coefficients multiplied by the corresponding vector group equals a zero vector, Then the vector group is linearly related.
- If the vector equation holds only when all coefficients are zero, that is, the set of vectors is linearly dependent only if each vector is a zero vector, then the set of vectors is linearly independent.
2. **Use the properties of rank to judge**:
- When the rank of a vector group is equal to the number of vectors, if the determinant is not zero, then the vector group is linearly independent; if the determinant is zero, then the vector group is linearly independent linearly related.
- When the rank of a vector group is less than the number of vectors, the vector group must be linearly related.
- For any matrix, if its rank is less than its column number, the corresponding column vector group is linearly related; if its rank is equal to the column number, it may be linearly related or linearly independent, and it needs to be further judged by the determinant.
3. **Judgment using homogeneous linear equations**:
- The necessary and sufficient condition for the linear correlation of vector groups is that the corresponding homogeneous linear equations have non-zero solutions.
- A necessary and sufficient condition for a system of vectors to be linearly independent is that the corresponding system of homogeneous linear equations has zero solutions.
4. **Use orthogonality to judge**:
- If the vectors in the vector group are orthogonal to each other, that is, the dot product of any two different vectors is zero, then the vector group is linearly independent.
- On the other hand, a group of vectors is linearly dependent if there are non-zero vectors in the group that can be linearly represented by other vectors.
5. **Use the maximum linearly independent group to judge**:
- If there is a maximum linearly independent group, then any vector in this group cannot be linearly represented by other vectors, and adding any other vectors will make the group linearly related.
6. **Special case judgment**:
- The vector group containing the zero vector must be linearly related, because the zero vector can be linearly represented by any vector.
- A set of vectors with only one non-zero vector is linearly independent because no vector can represent this unique non-zero vector.
In actual operation, the linear correlation of the vector group can be judged by constructing a coefficient matrix and calculating its rank, or directly calculating the determinant. For more complex cases, it may be necessary to use the row echelon form of the matrix or utilize linear algebra software tools to make the determination.