This doesn't mean anything to me. I believe it should be written ATx = 0. ATx is the product of A transpose and some vector x.negation said:What does ATx=0 means?
negation said:Does this notation means if A = [3,2;1,2;4,4], then, AT = [3,1,4;2,2,4]
and ATx [x1;x2;x3] = 0?
The nullspace of the transposed of the matrix A?
Mark44 said:This doesn't mean anything to me. I believe it should be written ATx = 0. ATx is the product of A transpose and some vector x.
My guess is they're asking whether the multiplication is defined. If x ##\in## R3, and A is as you have in post 1, then Ax is not defined, but ATx is defined.negation said:What significance is there if a question ask if it is consistent or inconsistent?
ATx = 0
The nullspace of A transpose x is the set of all vectors x that, when multiplied by the transpose of matrix A, result in a zero vector. This means that the nullspace contains all vectors that have no effect on the original matrix when multiplied by its transpose.
The nullspace of A transpose x is the orthogonal complement of the nullspace of A. This means that the two nullspaces are orthogonal to each other, and any vector in the nullspace of A transpose x is perpendicular to any vector in the nullspace of A.
The dimension of the nullspace of A transpose x is equal to the number of linearly independent columns of A. This is also known as the rank of A.
The nullspace of A transpose x can be calculated by finding the nullspace of A and then taking the orthogonal complement of that space. This can be done using techniques such as row reduction or the use of the Gram-Schmidt process.
The nullspace of A transpose x is important because it provides information about the linear independence of the columns of A. If the nullspace is non-empty, it means that there are linearly dependent columns in A. Additionally, the nullspace can be used to find solutions to systems of linear equations and to perform dimensionality reduction in data analysis.