Final answer:
To find an elementary matrix E that transforms vector a into b, identify the necessary row operations from a to b and construct E accordingly. E represents a specific transformation when multiplied with another matrix or vector.
Step-by-step explanation:
To find an elementary matrix E such that Ea = b, you are essentially searching for a matrix that can transform vector a into vector b when they are multiplied. The elementary matrix E applies a specific row operation to matrix A to achieve this transformation. To solve this equation, you'll need to consider the knowns and unknowns. In this case, matrix A and vector b are known, while the elementary matrix E is the unknown.
To proceed with the calculation, first write down the known vectors a and b, and then determine the type of elementary row operations needed to convert a into b. These operations can include row swapping, row multiplication, or row addition. Upon identifying the necessary operations, construct elementary matrix E. For example, if a is transformed into b simply by multiplying one of its rows by a constant c, then E would be an identity matrix with c substituted in the row corresponding to the operation.
It's important to note that elementary matrices are the building blocks of matrix operations. They are instrumental in achieving row echelon form or reduced row echelon form in matrices, which are essential steps in solving systems of linear equations or inverting matrices. Additionally, the relationship E(A) can be thought of as applying the transformation defined by elementary matrix E to every column of A.