In linear algebra, a set of vectors is linearly dependent if at least one of the vectors in the set can be written as a linear combination of the others. This means there exist scalars, not all zero, such that a linear combination of the vectors equals the zero vector. If no such non-trivial linear combination exists, the vectors are said to be linearly independent. Determining whether vectors are linearly dependent is a fundamental skill, essential for understanding concepts like basis, dimension, and eigenvalues.
Match the following terms with their correct definitions:
| Term | Definition |
|---|---|
| 1. Linear Combination | A. A set of vectors where at least one can be written as a sum of scalar multiples of the others. |
| 2. Linear Independence | B. An equation of the form $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ where not all $c_i$ are zero. |
| 3. Linear Dependence | C. A sum of scalar multiples of vectors. |
| 4. Trivial Solution | D. A set of vectors where the only solution to $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ is $c_1 = c_2 = ... = c_n = 0$. |
| 5. Non-Trivial Solution | E. The solution to a homogeneous equation where all variables equal zero. |
A set of vectors {$v_1, v_2, ..., v_n$} is considered linearly ________ if the only solution to the equation $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ is the ________ solution, where all scalars $c_i$ are equal to ________. Conversely, if there exists a set of scalars, not all ________, that satisfy this equation, the vectors are linearly ________.
Explain, in your own words, why understanding linear dependence is important in the context of finding a basis for a vector space. Give a practical example.
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