UPSC Exam > UPSC Videos > Mathematics Optional Notes for UPSC > Linear Dependence and Independence
Linear Dependence and Independence Video Lecture | Mathematics Optional Notes for UPSC
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FAQs on Linear Dependence and Independence Video Lecture - Mathematics Optional Notes for UPSC
| 1. What is the difference between linear dependence and linear independence? |  |
Ans. Linear dependence occurs when a set of vectors can be combined to form a nontrivial linear combination that equals the zero vector, while linear independence exists when no nontrivial linear combination of the vectors results in the zero vector.
| 2. How can we determine if a set of vectors is linearly independent? | |
Ans. One way to determine if a set of vectors is linearly independent is by forming a matrix with these vectors as columns and performing row reduction to check if the matrix has a pivot in every column.
| 3. Can a set of vectors be both linearly dependent and linearly independent? | |
Ans. No, a set of vectors cannot be both linearly dependent and linearly independent at the same time. It can only be one of the two.
| 4. What is the significance of linear independence in linear algebra? | |
Ans. Linear independence is crucial in linear algebra as it helps in determining the dimension of vector spaces, providing insight into the structure and properties of systems of linear equations.
| 5. How does linear dependence affect the solution of a system of linear equations? | |
Ans. In the context of solving systems of linear equations, linear dependence among vectors can lead to redundancy and ambiguity in the solutions, making it essential to identify and address such dependencies for accurate and meaningful results.
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