Final answer:
To find a basis, we put the vectors into a matrix and reduce it to its row echelon form, then take the non-zero rows. These form the basis and their count represents the dimension of the subspace. Vectors are represented in component form with unit vectors i, j, and k.
Step-by-step explanation:
To find a basis for the subspace spanned by the given vectors, we need to determine a set of vectors that are linearly independent and span the same space as the original set of vectors. This is often done by putting the vectors into a matrix and then reducing the matrix to its row echelon form or reduced row echelon form. The non-zero rows then represent a basis for the subspace. The dimension of the subspace is the number of vectors in the basis.
The vector component form in terms of the unit vectors of the axes involves breaking down a vector into its constituents along the x, y, and z axes, which are represented by i, j, and k respectively. For example, a vector G with scalar components a, b, and c would be written in vector component form as ai + bj + ck.