write a matlab function called find bases to find basis vectors for the column space and null space of a given matrix. in matlab

To write a MATLAB function called find_bases that finds basis vectors for the column space and null space of a given matrix, you can follow these steps:

main.m
function [col_space_basis, null_space_basis] = find_bases(matrix)
    % Find the basis for the column space
    col_space_basis = matrix(:, any(matrix));
    
    % Find the basis for the null space
    [~, ~, V] = svd(matrix);
    null_space_basis = V(:, end-size(matrix, 2) + 1 : end);
end
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Here's a breakdown of the code:

1. The function find_bases takes a matrix as an input and returns two outputs: the basis vectors for the column space (col_space_basis) and the basis vectors for the null space (null_space_basis).
2. The first line inside the function finds the basis for the column space using logical indexing. The any function is used to check if any of the columns have non-zero values. The resulting Boolean vector is used to select the corresponding columns of the input matrix.
3. The second line uses the singular value decomposition (svd) of the input matrix. The left singular vectors are stored in the matrix U, the singular values are stored in the diagonal matrix S, and the right singular vectors are stored in the matrix V.
4. The third line selects the last columns of the matrix V, corresponding to the null space of the matrix. The number of selected columns is equal to the number of singular values that are non-zero in the matrix S.
5. The calculated col_space_basis and null_space_basis are returned as the outputs of the function.

You can call this function in MATLAB by providing a matrix as an argument, for example:

main.m
matrix = [1 2 3; 4 5 6; 7 8 9];
[col_basis, null_basis] = find_bases(matrix);
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The variables col_basis and null_basis will then contain the basis vectors for the column space and null space, respectively, of the provided matrix.