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Linear Dependence & Independence Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
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FAQs on Linear Dependence & Independence Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is linear dependence and independence? |  |
Ans. Linear dependence refers to a situation where a vector can be expressed as a linear combination of other vectors in a given set. On the other hand, linear independence occurs when no vector in a set can be written as a linear combination of the others in that set.
| 2. How can we determine if a set of vectors is linearly dependent or independent? | |
Ans. To determine the linear dependence or independence of a set of vectors, we can consider the following condition: If the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 (where c1, c2, ..., cn are constants and v1, v2, ..., vn are the vectors in the set) is c1 = c2 = ... = cn = 0, then the set is linearly independent. Otherwise, it is linearly dependent.
| 3. Can a set of vectors be both linearly dependent and independent? | |
Ans. No, a set of vectors cannot be both linearly dependent and independent at the same time. It is either one or the other. If a set is linearly dependent, it means that at least one vector in the set can be expressed as a linear combination of the others. Conversely, if a set is linearly independent, no vector in the set can be written as a linear combination of the others.
| 4. Is the zero vector always included in a linearly dependent set of vectors? | |
Ans. Yes, the zero vector (0) is always included in a linearly dependent set of vectors. This is because the zero vector can always be expressed as a linear combination of any other vectors in the set by assigning all the scalar coefficients to be zero.
| 5. Can a set of two vectors be linearly dependent if one vector is a scalar multiple of the other? | |
Ans. Yes, a set of two vectors can be linearly dependent if one vector is a scalar multiple of the other. In such a case, one vector can be expressed as a scalar multiple of the other, indicating a linear dependence. For example, if v1 = 2v2, then the set {v1, v2} is linearly dependent.
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Nov 25, 2024 Last updated |
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