Find the basis of
M2={(a,b,c,d):a,b,c,d ∈ R}
To find the basis of the set M2, we need to first determine the linearly independent vectors that span the set.
The set M2 is defined as M2 = {(a, b, c, d) : a, b, c, d ∈ R}.
To determine the basis of M2, we need to find the vectors that span M2 and are linearly independent. In other words, we need to find a set of vectors such that any vector in M2 can be uniquely represented as a linear combination of these vectors.
Since M2 consists of all possible 4-tuples of real numbers, we can choose the standard basis vectors as a potential basis for M2. The standard basis vectors are defined as follows:
e1 = (1, 0, 0, 0),
e2 = (0, 1, 0, 0),
e3 = (0, 0, 1, 0),
e4 = (0, 0, 0, 1).
These vectors span the entire space of M2, as any vector in M2 can be written as a linear combination of these vectors.
Now, to determine if these vectors are linearly independent, we can form a matrix A using these vectors as columns:
A = [e1, e2, e3, e4] =
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
We can now row reduce the matrix A to row echelon form or reduced row echelon form. Since the matrix A is already in row echelon form, we can observe that the rank of A is 4, which is equal to the number of columns.
Since the rank of A is equal to the number of columns, we conclude that the vectors e1, e2, e3, and e4 are linearly independent.
Therefore, the basis of M2 is {e1, e2, e3, e4}, which consists of the standard basis vectors.