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, we need to understand the row space and column space of a matrix.
The row space of a matrix A, denoted as row(A), is the vector space spanned by the rows of matrix A. It represents the set of all linear combinations of the rows of A.
The column space of a matrix A, denoted as col(A), is the vector space spanned by the columns of matrix A. It represents the set of all linear combinations of the columns of A.
Now, let’s consider the transpose of matrix A, denoted as A^T. The transpose of A is obtained by interchanging its rows and columns.
If we take the row space of A^T (row(A^T)), we are essentially considering the vector space spanned by the rows of A^T, which is equivalent to the vector space spanned by the columns of A. In other words, row(A^T) = col(A).
Therefore, the row space of A^T is the same as the column space of A. Thus, the statement is true.
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