Question

g Question 11 pts A set of three vectors in R 4 could possibly be a linearly independent set of vectors. can never span R 4. can never be a basis for R 4. Flag this Question Question 21 pts A set of vectors in R 3 spans R 3 but is not linearly independent. How many vectors can the set have

Answer 1

Answer:

Step-by-step explanation:

We will solve this questions using some dimensionality analysis.

a) Note that $\mathbb{R}^4$ is a vector space of dimension 4. So, given a set of three vectors, they could be or could not be linearly independent. However, if they were, they would not span $\mathbb{R}^4$ since in order for us to span $\mathbb{R}^4$ we need 4 or more vectors. Since a set of three vector cannot span this space, they automatically cant be a basis for $\mathbb{R}^4$.

b)  $\mathbb{R}^3$ has a dimension 3. So, for a set of vectors A to span this space, we need that A contains at least 3 vectors that are linearly independent. So A could have possibly infinite number of vectors as long as there is a subset of A that is composed by 3 linearly independent vectors.