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iamzzz
- 22
- 0
Homework Statement
A is matrix m*n
show that nullspace of A is the subset of nullspace of A^T*A
You mean, you guess that AT*Ax is also 0, right? But why do you need to guess? If Ax = 0, you can easily prove AT*Ax = 0.iamzzz said:I know Ax=0 and i guess A^T*A is also 0
haruspex said:You mean, you guess that AT*Ax is also 0, right? But why do you need to guess? If Ax = 0, you can easily prove AT*Ax = 0.
iamzzz said:i love you guys thanks
The nullspace of a matrix A, also known as the kernel, is the set of all vectors that when multiplied by A result in the zero vector. In other words, it is the set of all solutions to the homogeneous equation Ax = 0.
The nullspace of A is a subset of the nullspace of A^T*A. This means that all vectors in the nullspace of A will also be in the nullspace of A^T*A, but there may be additional vectors in the nullspace of A^T*A that are not in the nullspace of A.
The dimension of the nullspace of A is also known as the nullity of A, and it is equal to the number of free variables in the reduced row echelon form of A. In other words, it is the number of linearly independent vectors in the nullspace of A.
The nullspace of A can be calculated by finding the reduced row echelon form of A and identifying the free variables. The free variables will correspond to the columns of the nullspace basis matrix, which can be multiplied by a scalar to generate all vectors in the nullspace of A.
Yes, it is possible for the nullspace of A to be empty. This will occur when the reduced row echelon form of A does not have any free variables, meaning that there are no linearly independent vectors that satisfy the homogeneous equation Ax = 0. This is also known as an inconsistent system of equations.