State with a brief justification whether the following state

State, with a brief justification, whether the following statements are true of false.
a. Every subset of three vectors in ^2 is linearly dependent.
b. Every subset of two vectors in ^2 is linearly independent.
c. The vectors (1, 2, 3) and (-1, 4, 6) span ^3 .
d. A single vector can be added to any two vectors in ^3 to get a basis for ^3 .
e. There exists a set that is linearly independent but does not span ^3 .

Please help me with this? Can you add a brief justification for each?

Solution

a) The given statement is TRUE, since we require only two basis vector for spanning R^2, hence the third vector will be depending on the other combination of two vectors

b) False, considering the two vectors i.e. [1 0] and [2 0], we can express the second vector as twice the first vector, hence the vectors can\'t be linearly independent

c) False,The two vectors doesn\'t span R^3, since we need atleast three vectors in 3-dimensional space to R^3,the basis vectors for R^3 is [1 0 0], [0 1 0] and [0 0 1]

d) False, let us suppose the single vector is [1 0 0] and the other two vectors are [1 1 0] and [2 0 0], the vectors won\'t span R^3, hence the given proposition is FALSE

e) The information is missing regarding how many vectors are there in the set, please check the problem