Question

Let {U1, U2, U3 } be a linearly dependent set of vectors. Select the best statement. A. {uj, U2, U3, U4} is a linearly independent set of vectors unless U4 is a linear combination of other vectors in the set. B. {uj, U2, U3, U4} could be a linearly independent or linearly dependent set of vectors depending on the vectors chosen. C. {uj, U2, U3, U4 } is always a linearly dependent set of vectors. D. {uj, U2, U3, U4 } is a linearly independent set of vectors unless U4 = 0. E. {uj, U2, U3, U4} is always a linearly independent set of vectors. F. none of the above

Answer 1

Option C is the correct statement. {uj, U2, U3, U4} is always a linearly dependent set of vectors.

Step-by-step explanation:

Linear dependence occurs in a set of vectors when one or more vectors can be expressed as a combination of others using scalar multiplication and addition. In other words, if the vectors in a set are linearly dependent, one can be written as a linear combination of the others.

The statement that best describes the linear dependence of the set {U1, U2, U3} is option C: {uj, U2, U3, U4} is always a linearly dependent set of vectors. This is because the original set {U1, U2, U3} is linearly dependent, which means that one or more of the vectors can be written as a linear combination of the others. When we add another vector U4 to the set, it is guaranteed to be linearly dependent on the original vectors, making the entire set linearly dependent.