Question

In this assignment, we will explore four subspaces that are connected 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 a linear transformation from R n to R m. Consider the following definitions:

• The transpose of A, denoted A^T is the n × m matrix we 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 the columns of A. col(A) is a subspace of R m and is the same as the image of T.

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

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

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

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

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

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

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

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

Answer 1

Q3 Null(A) is obtained by solving Ax=0, This mean that every row vector is perpendicular to every solution of Ax=0. To elaborate it, the dot product of every row vector of A and every vector x would be 0 only when the angle between the two is 90 degrees. This shows that row(A) and Null(A) orthogonal complements.

Q4. Null(A^T) would be obtained by solving A^Ty=0. This means that every row vector of A^Tis perpendicular to every solution of A^T y=0. By the same logic as given above, since the dot product A^T y is 0, every y vector will be perpendiular to every row vector of A^T . Now row vectors of A^T are the column vectors of A, hence it can be concluded that col(A) and Null(A^T ) are orthogonal complements.

Q5 Since A is mxn matrix , it has m rows and n columns. Column space would be in R^m . Since rank of A is r, it would have r pivot columns. Hence dimension of col(A) would be r. Since A has n columns, the dimension of Null(A) would be n-r.

Next, A^T would be a nxm matrix, it would have n rows and m columns. The row space of A, row(A) would be the col(A^T). Since the rank of A is r, rank of A^T would also be r. This means that it will a r pivot columns. Dimension of col(A^T) would be r. This means , the dimension of row(A)=r. Since there are m columns in A^T, the dimension of null(A^T) will be m-r.