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Understanding the column space of a matrix is fundamental in the world of linear algebra. It represents all the possible linear combinations of the matrix's columns. To visually grasp this concept, imagine each column of a matrix as an arrow in space. The area or the 'space' they cover when you move these arrows around, without changing their direction or length, is the column space.

For instance, let's consider the matrix from our exercise given by: \[ \left[\begin{array}{lll} 1 & 2 & 3 \ 4 & 5 & 6 \end{array}\right] \] Here, the column space is the set of all vectors that can be written as a linear combination of the matrix's columns. When seeking a basis for this space, we aim to find the fewest number of columns that still describe this area entirely. This directly leads to the inspection of linear dependence among the columns - a key step to determining the basis, as demonstrated in the provided step-by-step solution.

The term linear independence is used to describe a set of vectors where no vector in the set can be written as a combination of the others. If you think of vectors as building blocks, a set is linearly independent if you can't use these blocks to build each other. Linear independence is essential when finding a basis because a basis must be made of vectors that uniquely contribute to the space.

In our matrix exercise, we examined the vectors c1, c2, and c3. The realization that c3 could be constructed from c1 and c2 (c3 = c1 + 2*c2) indicated that c3 was not independent of c1 and c2. Consequently, c3 was unnecessary for our basis, and the basis of the image consisted only of the linearly independent vectors c1 and c2. Remember, a basis with any linearly dependent vector would be redundant, and such redundancy is not allowed in a minimal, efficient basis.

The span of a set of vectors can be imagined as the result of 'casting a net' with the vectors as the ropes. Wherever the net can reach by adjusting the length of the ropes, without adding new ones or changing their direction, is considered the span of these vectors. In mathematical terms, it is the collection of all vectors that can be formed by scaling and combining the original vectors.

In relation to our matrix, the span of the columns would encompass all possible vectors in a two-dimensional plane (since our matrix has two rows) that can be reached by using the columns as directions. By identifying that the vector c3 does not extend our net beyond what c1 and c2 already cover, we dismiss it from the spanning set. The remaining vectors c1 and c2 provide the maximum extent of reach or span for our 'net', making them the perfect candidates for the basis of the column space.