The Beauty of Row Space and Column Space in a 2x2 Matrix

What are the two different bases for a 2x2 matrix?

a) Column space, null space

b) Row space, column space

c) Null space, eigen space

d) Row space, eigen space

Answer:

The row space and column space are two different bases for a 2x2 matrix.

The correct answer is Row space, column space. In a 2x2 matrix, the row space refers to the vector space spanned by the rows of the matrix, while the column space refers to the vector space spanned by the columns of the matrix. These two spaces can be different from each other.

For example, consider the following matrix:

1 2

3 4

The row space is spanned by the row vectors (1, 2) and (3, 4), while the column space is spanned by the column vectors (1, 3) and (2, 4).