The only real difference between the spaces R 6 and M 2x3( R) is in the notation: The six entries denoting an element in R 6 are written as a single row (or column), while the six entries denoting an element in M 2x3( R) are written in two rows of three entries each. One consequence of this structural identity is that under the mapping ϕ-the isomorphism-each basis “vector” E igiven above for M 2x3( R) corresponds to the standard basis vector e ifor R 6. The conclusion is that the spaces M 2x3( R) and R 6 are structurally identical, that is, isomorphic, a fact which is denoted M 2x3( R) ≅ R 6. Is compatible with the vector space operations of addition and scalar multiplication. This one‐to‐one correspondence between M 2x3( R) and R 6, Then, to every matrix in M 2x3( R) there corresponds a unique vector in R 6, and vice versa. The rule here is simple: Given a 2 by 3 matrix, form a 6‐vector by writing the entries in the first row of the matrix followed by the entries in the second row. If the entries in a given 2 by 3 matrix are written out in a single row (or column), the result is a vector in R 6. (Alternatively, the only way k 1 E 1 + k 2 E 2 + k 3 E 3 + k 4 E 4 + k 5 E 5 + k 6 E 6 will give the 2 by 3 zero matrix is if each scalar coefficient, k i, in this combination is zero.) These six “vectors” therefore form a basis for M 2x3( R), so dim M 2x3( R) = 6. Furthermore, these “vectors” are linearly independent: none of these matrices is a linear combination of the others. Since M 2x3( R) is a vector space, what is its dimension? First, note that any 2 by 3 matrix is a unique linear combination of the following six matrices:
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