In this assignment, we will explore four subspaces that areconnected to a linear transformation. For the questions below,assume A is an m×n matrix with rank r, so that T(~x) = A~x is alinear transformation from R n to R m. Consider the followingdefinitions:

• The transpose of A, denoted A^T is the n × m matrixwe get from A by switching the roles of rows and columns – that is,the rows of A^T are the columns of A, and vice versa.

• The column space of A, denoted col(A), is the span of thecolumns of A. col(A) is a subspace of R m and is the same as theimage of T.

• The row space of A, denoted row (A), is the span of the rowsof A. row (A) is a subspace of Rⁿ .

• The null space of A, denoted null(A), is the subspace ofRⁿ made up of vectors x such that Ax = 0 and is equal tothe kernel of T.

• The left null space of A, denoted null( A^T) , isthe subspace of R^m made up of vectors y such thatA^Ty = 0.

col(A), row (A), null(A), and null (A^T ) are calledthe four fundamental subspaces for A.

We showed in class that the pivot columns of A form a basis forcol(A), and that the vectors in the general solution to Ax = 0 forma basis for null(A). Likewise, the vectors in the general solutionto A^Ty = 0 form a basis for null (A^T ).

Q3: Show that row (A) and null(A) are orthogonalcomplements.

Q4: Show that col(A) and null ( A^T ) are orthogonalcomplements.

Q5: Assuming A is an m × n matrix with rank r, what are thedimensions of the four subspaces for A?