The column space of A'*A is the set of all possible linear combinations of the columns of A'*A. In other words, it is the set of all possible vectors that can be obtained by multiplying A'*A by a scalar and adding the resulting vectors together.
The column space of A'*A is a subset of the column space of A. This means that every vector in the column space of A'*A can also be obtained by multiplying A by a scalar and adding the resulting vectors together. However, the column space of A'*A may also contain additional vectors that are not in the column space of A.
The dimension of the column space of A'*A is equal to the rank of A'*A. This can be found by performing row reduction on A'*A and counting the number of non-zero rows in the reduced matrix. The number of non-zero rows is the same as the number of pivot columns, which is equal to the rank.
Yes, the column space of A'*A is always a subspace of the original vector space. This is because it contains all possible linear combinations of the columns of A'*A, which is a fundamental property of subspaces.
Yes, the column space of A'*A can be larger than the column space of A. This can happen when the columns of A are not linearly independent, meaning that some of the columns can be written as a linear combination of the others. In this case, the column space of A will have a lower dimension than the column space of A'*A.