Question

Non empty subsets of linearly independent sets are linearly independent sets 6. Non empty subsets of linearly dependent sets are linearly dependent 7. Suppose U and Vare subsets of a vector space W, and U is contained in V. If U is linearly dependent, then Vis linearly dependent 8. Suppose U and Vare subsets of a vector space W, and U is contained in V. If U is linearly independent, then Vis linearly independent 9. Suppose U and Vare linearly dependent subsets of a vector space W. The union of U and Vis linearly dependent. 0. Suppose U and Vare linearly independent subsets of a vector space W. The union of U and V is linearly independent.

Answer 1

In linear algebra, non-empty subsets of linearly independent sets are also linearly independent, while non-empty subsets of linearly dependent sets are also linearly dependent. If a subset U is contained in a subset V, and U is linearly dependent, then V is linearly dependent. Conversely, if U is linearly independent, then V is also linearly independent. Moreover, if both U and V are linearly dependent or both U and V are linearly independent, their union will also be linearly dependent or linearly independent, respectively.

Step-by-step explanation:

In linear algebra, a subset of a vector space is linearly independent if no element in the subset can be expressed as a linear combination of other elements in the subset. Therefore, if we take a non-empty subset of a linearly independent set, it will also be a linearly independent set.

On the other hand, a subset of a linearly dependent set will also be linearly dependent. This is because if the original set is linearly dependent, it means that there exists at least one vector in the set that can be expressed as a linear combination of other vectors in the set. Therefore, any subset that includes this vector will also be linearly dependent.

Based on these concepts, we can conclude that if a subset U is contained in a subset V and U is linearly dependent, then V is also linearly dependent. Conversely, if U is linearly independent, then V is also linearly independent. Furthermore, if both U and V are linearly dependent, their union will also be linearly dependent. Similarly, if both U and V are linearly independent, their union will also be linearly independent.es this vector will also be linearly dependent.

Based on these concepts, we can conclude that if a subset U is contained in a subset V and U is linearly dependent, then V is also linearly dependent. Conversely, if U is linearly independent, then V is also linearly independent. Furthermore, if both U and V are linearly dependent, their union will also be linearly dependent. Similarly, if both U and V are linearly independent, their union will also be linearly independent.