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If v1, v2, …., v6 are six vectors in R4, which one of the following statements is FALSE?

If v1, v2, …., v6 are six vectors in R4, which one of the following statements is FALSE? Correct Answer Any four of these vectors form a basis for R<sup>4</sup>

Concept:

Basis: it is defined as a subset of vectors within the space that are linearly independent i.e. it can’t be defined as a set of a linear combination of any other vectors.

Span: It is defined as the linear combination of linearly independent vectors within the space.

R4: It shows the space of 4 linearly independent vectors.

Analysis:

1) R4 means that there are only 4 linearly independent vectors out of 6 vectors. So it is not necessary that these vectors span R4. (TRUE STATEMENT)

2) All these vectors are not linearly independent because there are only 4 linearly independent vectors out of 6 vectors. (TRUE STATEMENT)

3) Any four of these vectors can’t form a basis for R4 because only linearly independent vectors can form the basis for R4. Out of 6 vectors, only 4 vectors are linearly independent. So, this statement is wrong. (WRONG STATEMENT)

4) If {v1, v3, v5, v6} span R4, it means that vectors {v1, v3, v5, v6} are linearly independent, because span shows the linear combination of linearly independent vectors. Hence, it forms a basis for R4. (TRUE STATEMENT)

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