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

To prove that the set of vectors (1, 2), (3, 4) is linearly independent and spanning in R2, we need to follow these steps:

Step 1: Linear Independence We need to show that the only solution to the equation a*(1, 2) + b*(3, 4) = (0, 0) is a = 0 and b = 0.

This gives us the system of equations: 1a + 3b = 0 2a + 4b = 0

This system has only the trivial solution a = 0, b = 0. Therefore, the vectors are linearly independent.

Step 2: Spanning R2 To show that these vectors span R2, we need to show that any vector in R2 can be written as a linear combination of these vectors.

Let's take an arbitrary vector (x, y) in R2. We need to find scalars a and b such that: a*(1, 2) + b*(3, 4) = (x, y)

This gives us the system of equations: a + 3b = x 2a + 4b = y

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

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