Nullspace(kernel) and transpose

[complexhuman]

hmmm...I have problems understanding this...how can the null space if a matrix(not necessarily a square) be the same as that of its transpose?

Thanks in advance

 

[AKG]

If the matrix is not square, then this is impossible. The null space of a matrix A consists of vectors x such that Ax = 0. If A is not square, and Ax is defined (i.e. you are allowed to multiply A and x) then A^Tx is not even defined. I'm not sure what you're asking though. In general, the null space of a matrix is not the same if it as the null space of its transpose. However, certainly if the matrix is symmetric then its kernel is the same as the kernel of its transpose, since the matrix is its own transpose.

 

[tacman]

The null space of a matrix, also known as the kernel, is the set of all vectors that, when multiplied by the matrix, result in a zero vector. This means that the null space contains all the vectors that are mapped to the origin by the matrix.

The transpose of a matrix is obtained by interchanging the rows and columns of the original matrix. This means that the transpose of a matrix has the same number of rows and columns as the original matrix, but they are arranged in a different way.

Now, it may seem counterintuitive that the null space of a matrix and its transpose can be the same. However, this is actually possible because the null space is not dependent on the arrangement of rows and columns in a matrix. It is solely determined by the values in the matrix.

For example, let's say we have a 3x2 matrix A and its transpose A^T. The null space of A would contain all the vectors that, when multiplied by A, result in a zero vector. Similarly, the null space of A^T would contain all the vectors that, when multiplied by A^T, result in a zero vector. Since A and A^T have the same values, the same vectors would result in a zero vector when multiplied by either of them. Therefore, the null space of A and A^T would be the same.

In other words, the null space of a matrix is a property of the matrix itself, not its arrangement of rows and columns. So, even though the transpose of a matrix may look different, it can still have the same null space as the original matrix.