Suppose we want to find the projection p of b onto V spana1

Suppose we want to find the projection p of b onto V = span{a1, a2, . . . , an} where the ai’s are linearly dependent. Why will our usual formula p = A(A^TA)^-1A^Tb fail? What can we do to get around this problem and still find p?

Solution

Since the vectors a1,a2,...,an are linearly dependent, hence the matrix A will not be having an inverse since the determinant of |A| will be equal to zero

Therefore, we cannot find p since the inverse doesn\'t exists for (A^TA), hence in order to get around this problem, we need to have the vectors a1,a2,..,an as linearly independent and in that case we can find the A^- since |A| will not be equal to zero