Let's consider two sets in the linear algebra context. Set A contains the vectors [1, 2], [3, 4] and set B contains the vectors [1, 2], [3, 4], [5, 6]. Is set A a proper subset of set B?

Step 1 ：First we need to understand what a proper subset is. A is a proper subset of B if every element of A is also an element of B but B contains at least one element that A does not.

Step 2 ：Now we analyze the two sets. Set A has the vectors [1, 2] and [3, 4], which are also in set B. However, set B also contains the vector [5, 6], which is not in set A.

Step 3 ：Therefore, set A is a proper subset of set B because it meets the criteria.