11. Let both of and be 3-dimensional column vectors , represented as follows, 12. Where the vector \ mathbf { 1 } is a column vector of ones. 13. Here is thought of as a column vector containing components with the allowed values of. 14. In components, such operator is expressed with orthogonal matrix that is multiplied to column vectors . 15. Again, is thought of as a column vector containing components with the allowed values of. 16. This matrix must be the product of a single column vector with a single row vector. 17. This produces a basis for the column space that is a subset of the original column vectors . 18. Another important example is the transpose operation in linear algebra which takes row vectors to column vectors . 19. In this case, our data vector, d is a column vector of dimension ( 5x1 ). 20. Here, \ mathbf { 1 } is a column vector of 1's of dimension M.