Let A be upper triangular and tridiagonal If A is n times n

Let A be upper triangular and tridiagonal. If A is n times n, show that the number of floating point operations required to solve Ax = b is 3n - 2.

Solution

Ans-

Vector Representation Relative to a Basis

Suppose that Vis a vector space and B={v1,v2,v3,…,vm} is a linearly independent set that spans V. Let w be any vector in V. Then there exist unique scalars a1,a2,a3,…,am such that

w=a1v1+a2v2+a3v3++amvm.

Proof

The converse of Theorem VRRB is true as well, but is not important enough to rise beyond an exercise (see Exercise LISS.T51).

This is a very typical use of the hypothesis that a set is linearly independent — obtain a relation of linear dependence and then conclude that the scalars must all be zero. The result of this theorem tells us that we can write any vector in a vector space as a linear combination of the vectors in a linearly independent spanning set, but only just. There is only enough raw material in the spanning set to write each vector one way as a linear combination. So in this sense, we could call a linearly independent spanning set a “minimal spanning set.” These sets are so important that we will give them a simpler name (“basis”) and explore their properties further in the next section.