Final answer:
The smallest possible rank of a 4×7 matrix is 2, which means that there are two linearly independent rows or columns, and all others can be expressed as linear combinations of these two. This reflects the minimal dimension of vector space spanned by the matrix's rows or columns.
Step-by-step explanation:
The smallest possible rank of a 4×7 matrix refers to the minimum number of linearly independent rows or columns that the matrix can have. The rank of a matrix is also the dimension of the vector space spanned by its rows or columns. For a 4×7 matrix, since there are only 4 rows, the maximum possible rank is 4, because you cannot have more linearly independent rows than the number of rows itself.
The smallest possible rank would be the situation in which the minimal number of these rows or columns are linearly independent. In mathematical terms, if all rows or columns can be expressed as a linear combination of just two rows or columns, the rank would be 2. This means that even though the matrix is of a larger size, the actual 'information' it contains is only of the dimension of 2, since all other rows or columns can be generated from these two linearly independent ones.
To summarize, a 4×7 matrix could have a rank as low as 2 if the conditions above are met. This is under the assumption that at least two rows or columns are linearly independent, and every other row or column is a linear combination of these two.