Linearly dependent vectors are vectors in a vector space that can be expressed as a linear combination of one or more of the other vectors in the set. In simpler terms, there exists a non-trivial combination of these vectors that results in the zero vector.

Definition:

A set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ in a vector space is linearly dependent if there exist scalars $c_1, c_2, \dots, c_n$, not all zero, such that: $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n = \mathbf{0}.$

Here:

• $c_1, c_2, \dots, c_n$ are the coefficients of the linear combination.
• At least one coefficient must be non-zero.

If no such non-trivial combination exists, the vectors are linearly independent.

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Key Characteristics of Linear Dependence:

1. Redundancy: At least one vector in the set can be written as a combination of others.
2. Dimension Connection: In $\mathbb{R}^n$, any set of more than $n$ vectors is always linearly dependent.
3. Determinant Test: For $n$ vectors in $n$-dimensional space, the set is dependent if the determinant of the matrix formed by placing the vectors as columns is zero.
4. Zero Vector Inclusion: If the set includes the zero vector, the set is automatically linearly dependent.

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Example:

Consider the vectors $\mathbf{v}_1 = (1, 2, 3)$, $\mathbf{v}_2 = (2, 4, 6)$, and $\mathbf{v}_3 = (0, 1, 1)$ in $\mathbb{R}^3$. They are linearly dependent because: $2\mathbf{v}_1 - \mathbf{v}_2 = \mathbf{0}.$ Here, $c_1 = 2$, $c_2 = -1$, and $c_3 = 0$ (not all zero).

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Tip:

To test if vectors are linearly dependent, place them as columns of a matrix and calculate the determinant (if square) or reduce the matrix to row-echelon form to check for a dependency.

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Would you like examples or further details on determining linear dependence? Here are some related questions to expand understanding:

1. How do linearly dependent vectors relate to basis vectors in a vector space?
2. How can the rank of a matrix help identify linear dependence?
3. Can linearly dependent vectors form a spanning set of a vector space?
4. How do linearly dependent vectors differ in infinite-dimensional spaces?
5. What is the role of linear dependence in solving systems of equations?

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