What does the term "column space" refer to in relation to a matrix?

The term "column space" refers to the span of the column vectors of a matrix. This means that the column space is the set of all possible linear combinations of the matrix's column vectors. It is a fundamental concept in linear algebra, as it describes the range of outputs that can be obtained from the matrix when it is used to transform different input vectors.

Understanding the column space is crucial for solving linear systems, as it provides insight into the solutions' existence and uniqueness. If a vector lies within the column space, it means that the vector can be expressed as a linear combination of the matrix's columns, signifying that the corresponding linear system has at least one solution. The dimension of the column space, known as the rank of the matrix, indicates how many of the columns are linearly independent, thus influencing the behavior of the linear transformations represented by the matrix.

The other options pertain to different concepts within linear algebra. The set of all possible row vectors is related to the row space, while the determinant provides information about the matrix's invertibility and the volume scaling factor of linear transformations. The notion of linear independence is important, but it specifically deals with the relationships among columns rather than defining the column space itself. Thus, the correct