Is the Row Space of the Transpose of a Matrix the Same as the Column Space of the Original Matrix? A Proof.

A denotes an m×n matrix. Determine whether the statement is true or false. Justify your answer.The row space of AT is the same as the column space of A.

The statement is true

The statement is true.

To justify this, let’s first define the row space and column space of a matrix:

– The row space of a matrix A is the subspace spanned by the rows of A.
– The column space of a matrix A is the subspace spanned by the columns of A.

Now, let’s consider the transpose of matrix A, denoted as AT. The dimensions of AT will be n×m, as the rows and columns of A are interchanged.

The row space of AT will be the subspace spanned by the rows of AT. Since the rows of AT are originally the columns of A, the row space of AT will be the subspace spanned by the columns of A, which is the column space of A.

Thus, the row space of AT is indeed the same as the column space of A.

In summary, the statement is true because the row space of AT is equal to the column space of A.

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