Prove that the set of vectors (1, 2), (3, 4) is linearly independent and spanning in R2

Sure, let's go through this step by step.

1. Linear Independence: Two vectors are linearly independent if the only solution to the equation c1v1 + c2v2 = 0 is c1 = c2 = 0. Here, v1 = (1, 2) and v2 = (3, 4). So, we need to solve the following system of equations:

   c11 + c23 = 0 c12 + c24 = 0

   This system of equations has only the trivial solution c1 = c2 = 0. Therefore, the vectors (1, 2) and (3, 4) are linearly independent.

2. Spanning: A set of vectors spans R2 if any vector in R2 can be written as a linear combination of the vectors in the set. In other words, for any vector (a, b) in R2, there exist scalars c1 and c2 such that c1v1 + c2v2 = (a, b).

   Let's take an arbitrary vector (a, b) in R2. We need to solve the following system of equations:

   c11 + c23 = a c12 + c24 = b

   This system of equations has a solution for any (a, b), which means any vector in R2 can be written as a linear combination of (1, 2) and (3, 4). Therefore, the vectors (1, 2) and (3, 4) span R2.

So, the set of vectors (1, 2) and (3, 4) is linearly independent and spanning in R2.