Question

Let A be an m×n matrix. We write programs in Matlab to find the 4 fundamental subspaces of A. (a) Write a function with the prototype B=columnSpace (A), where the input is the given matrix A and the output, B, is a matrix whose columns form a basis for the column space of A.

Answer 1

The question is about writing a Matlab function to find a basis for the column space of a matrix A. The function uses Matlab's rref function to identify the pivot columns which are used to construct the output matrix B, containing a basis for the column space.

Step-by-step explanation:

The student has asked for a Matlab function that can find a basis for the column space of a matrix A. In linear algebra, the column space of a matrix is the span (all possible linear combinations) of its column vectors. To find a basis for the column space, we can perform column operations to reduce the matrix to its echelon form or its reduced echelon form. In Matlab, this process can be easily done using the rank function and logical indexing.

Matlab Function for Column Space

Here is a Matlab function that accomplishes this:

function B = columnSpace(A)
[~,~,pivcol] = rref(A);
B = A(:,pivcol);
end

The function uses Matlab's rref function to compute the reduced row echelon form of A and returns the pivot columns. These pivot columns form a basis for the column space of A. By selecting the columns from the original matrix A that correspond to the pivot columns from rref, the function columnSpace ensures that the original structure of the column space is maintained in the output matrix B.