Find the basis of
M2={(a,b,c,d):a,b,c,d ∈ R}
To find the basis of a set, we need to find a set of vectors that spans the set and is linearly independent.
In this case, M2 is defined as the set of all (a, b, c, d) where a, b, c, d are real numbers. To determine the basis of M2, we need to find a set of vectors that can represent all possible combinations of (a, b, c, d) and are linearly independent.
One possible basis for M2 is the set of standard unit vectors: {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}.
To show that this set of vectors is a basis, we need to demonstrate that it spans M2 and is linearly independent.
1. Spanning: Any vector (a, b, c, d) in M2 can be represented as a combination of these standard unit vectors. For example, (a, b, c, d) = a(1, 0, 0, 0) + b(0, 1, 0, 0) + c(0, 0, 1, 0) + d(0, 0, 0, 1). Thus, the set of standard unit vectors spans M2.
2. Linear Independence: We can show that the set of standard unit vectors is linearly independent by setting up the following equation:
x(1, 0, 0, 0) + y(0, 1, 0, 0) + z(0, 0, 1, 0) + w(0, 0, 0, 1) = (0, 0, 0, 0)
Setting up the resulting equations, we get:
x = 0
y = 0
z = 0
w = 0
The only solution to this system of equations is x = y = z = w = 0, indicating that the set of standard unit vectors is linearly independent.
Therefore, the set of standard unit vectors {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} forms a basis for M2.