Question

For each of the situations described below,give an example(if it’s possible) or explain why it’s not possible.

Answer 1

Step-by-step explanation:

a) If the given set of vectors does not span $\mathbb{R}^{3}$ , it means the number of linearly independent vectors are less than 3. So by adding one or more linearly independent vectors with respect to existing vectors, we can convert the set to basis and basis always spans vector space.

eg. $S= \left \{ (1,0,0)^{T},(1,2,0)^{T} \right \}$ this set does not span $\mathbb{R}^{3}$ . Since it has only two vectors and both vectors are linearly independent, so adding one linearly independent vector with respect to the vectors of S viz. (0,0,1)^{T} can span whole $\mathbb{R}^{3}$.

b) It is not possible. Since if a set of vectors is linearly dependent then after adding linearly independent vectors it will also linearly dependent i.e. addition of a linearly independent vector does not effect on the set.