Question

give an example of two linearly dependent sets in a vector space whose intersection is linearly independent​

Answer 1

Given:

Two linearly dependent sets in a vector space whose intersection is linearly independent​.

To find:

Give an example of two linearly dependent sets in a vector space whose intersection is linearly independent​.

Solution:

From given statement, let us consider two linearly dependent sets,

B1 and B2 given by,

B1 = {(1, 0), (0, 1), (1, 2)} and B2 = {(1, 0),(0, 1),(1, 1)}

B1 and B2 are linearly dependent set in R² as the dimension of R² over R is 2 and B1 and B2 have three vectors, so they have to be dependent.

The intersection of these is given by,

B1 ∩ B2 = {(1, 0), (0, 1)} → linearly independent in R².

The intersection of B1 and B2 contains one vector in x-axis direction and another in y-axis direction.

So the intersection is an independent set of vectors.