Linear dependence

Definition

Linear dependence occurs when a set of vectors can be expressed as a linear combination of other vectors in the set, meaning at least one vector in the set can be written as a combination of the others. This concept is crucial for understanding the properties of vector spaces and directly relates to the evaluation of determinants, as a set of linearly dependent vectors leads to a determinant value of zero, indicating that the vectors do not span the space.

5 Must Know Facts For Your Next Test

1. If a set of vectors is linearly dependent, at least one vector can be written as a linear combination of others, indicating redundancy in the set.
2. The determinant of a matrix formed by linearly dependent vectors is zero, which means that these vectors do not occupy full dimensional space.
3. A square matrix is invertible if and only if its columns are linearly independent, which is directly linked to having a non-zero determinant.
4. In an n-dimensional space, any set of more than n vectors must be linearly dependent due to the pigeonhole principle.
5. Linear dependence can be checked using methods such as row reduction or calculating determinants for small matrices.

Review Questions

• How does linear dependence affect the determinant of a matrix, and what does this imply about the vectors represented by the matrix?
  • When vectors in a matrix are linearly dependent, the determinant of that matrix is zero. This indicates that these vectors do not span the entire space they occupy, meaning they cannot represent unique directions in that dimensionality. Consequently, it suggests that at least one vector can be expressed as a combination of others, leading to redundancy in information represented by the matrix.
• Discuss how you would determine if a given set of vectors is linearly dependent or independent using row reduction.
  • To determine if a set of vectors is linearly dependent or independent using row reduction, you first construct a matrix with these vectors as its columns. By performing Gaussian elimination or row echelon form on this matrix, if you find any row reduced to all zeros, it indicates that there is at least one linear relationship among the vectors. If every column has a leading entry, the vectors are independent; otherwise, they are dependent.
• Evaluate the implications of having a linearly dependent set of vectors on the dimensions and properties of vector spaces they inhabit.
  • Having a linearly dependent set of vectors implies that the dimension of the vector space spanned by these vectors is less than the number of vectors present. This means some vectors do not contribute new directions within that space and are essentially redundant. In practical terms, this can limit the ability to solve systems of equations uniquely since having more variables than independent equations often leads to infinite solutions or no solutions at all.

Related terms

linear independence: A condition where no vector in a set can be written as a linear combination of the others, indicating that each vector adds unique direction in space.

span: The set of all possible linear combinations of a given set of vectors, representing the space that those vectors can fill.

rank: The maximum number of linearly independent column vectors in a matrix, which reflects the dimension of the column space.