Question

If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. اختر احدى الدجابات True False

Answer 1

If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.

Step-by-step explanation:

A set containing fewer vectors than there are entries in the vectors is always linearly independent if and only if none of the vectors can be written as a linear combination of the others.

To show that a set is linearly independent, we need to assume that the vectors in the set are linearly dependent and then show that this leads to contradiction.

For example, consider the set {v1, v2, v3}, where v1, v2, and v3 are vectors in three-dimensional space. If the set contains two vectors, say v1 and v2, and v1 and v2 are not scalar multiples of each other, then the set is linearly independent.

Answer 2

If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent: False.

In Mathematics and Euclidean Geometry, a vector typically comprises two points. First, is the starting point which is commonly referred to as the "tail" and the second (ending) point that is commonly referred to the "head."

Generally speaking, there exist a set which consists of fewer vectors than there are entries in the vectors, that is linearly dependent, rather than being independent. For exampe, a set that is composed of two (2) vectors in which one of the vectors is a scalar multiple of the other vector.

Additionally, the following set of the single vector
`\left[\begin{array}{ccc}0\\0\end{array}\right]` is made up of two (2) entries, but it is not linearly independent.

In conclusion, we can logically deduce that given sentence is a false statement.