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iamzzz
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Why is that Nullspace of A is subset of nullspace of A^T*A
let's say that A is m*n matrix
let's say that A is m*n matrix
The nullspace of a matrix A is the set of all vectors that when multiplied by A result in the zero vector. Therefore, any vector in the nullspace of A is also in the nullspace of A^T*A because multiplying by A^T*A is equivalent to multiplying by A and then by A^T. This means that any vector that was already in the nullspace of A will remain in the nullspace of A^T*A, making it a subset.
The transpose of a matrix A switches its rows and columns. This means that the nullspace of A^T is composed of the vectors that are orthogonal to the rows of A. Furthermore, multiplying by A^T*A results in a square matrix, which means that the nullspace of A^T*A is the set of all vectors that are orthogonal to the rows of A^T, which are the same as the columns of A. Therefore, the nullspace of A^T*A is a subset of the nullspace of A.
Yes, it is possible for the nullspace of A and the nullspace of A^T*A to be equal. This can happen if A is a square matrix with linearly independent columns, meaning that it has a trivial nullspace (only the zero vector). In this case, both A and A^T*A will have the same nullspace, which is just the zero vector.
This property has important implications in linear algebra and matrix operations. It means that if a vector x is in the nullspace of A, then it is also in the nullspace of A^T*A. This allows us to simplify calculations and reduce the dimensionality of the problem when solving linear systems of equations involving A^T*A.
No, the nullspace of A is not always a subset of the nullspace of A^T*A. This property only holds when A is a rectangular matrix with linearly independent columns. If A is a singular matrix (not invertible) or has linearly dependent columns, then the nullspace of A will not be a subset of the nullspace of A^T*A. In fact, the nullspace of A^T*A may be larger than the nullspace of A in these cases.