Ax is in the column space (col(A)) of matrix A because it is a linear combination of the columns of A. This means that Ax can be expressed as a combination of the columns of A, and therefore is contained within the column space of A.
In this equation, (A^T) represents the transpose of matrix A. When multiplied with matrix A, the result is a square matrix. The product of a matrix and its transpose is always a symmetric matrix, which means that it is equal to its own transpose. Since the product (A^T)Ax is equal to its transpose, it is also equal to 0.
When (A^T)Ax is equal to 0, it means that the vector Ax is orthogonal, or perpendicular, to all of the columns of A. This is because the dot product of two orthogonal vectors is equal to 0. In other words, Ax is perpendicular to each column of A, meaning it lies in the null space of A^T.
This equation demonstrates the relationship between the null space and column space of A. The null space of A^T is the set of all vectors that satisfy (A^T)x=0, which means that Ax is contained within this null space. At the same time, Ax is also in the column space of A, as explained in the answer to question 1. This shows that the null space and column space of A are not mutually exclusive, and Ax can exist in both of these spaces simultaneously.
The equation (A^T)Ax=0 is important in linear algebra because it provides insight into the properties and relationships of matrix operations. It also serves as a key step in solving systems of linear equations and finding the null space and column space of a matrix. Additionally, this equation is used in various applications, such as image processing and data compression, to analyze and manipulate data efficiently.