How do you show whether or not a vector is a linear combination of a set of vectors?
There is no way to make the last vector as a linear combination of the first 5. Put the first 5 vectors as column vectors in a matrix “A”. Find the rank(A) =3; this means that only 3 of the 5 are linearly independent. The rank can be found using calculator, software, or by using Gaussian elimination and forming an upper triangular matrix and counting the number of pivot elements. The notation you are using looks like MatLab, so use the command “rank”. Let’s call that last vector, [0; -1; 10; -8; 7; 7], “b”. Then you are looking for a solution “x” to the system Ax=b If such a solution exists, you can make “b” as a linear combination of the columns of “A”. No solution exists. You can show this by augmenting “b” to “A” making a 6×6 matrix “C”. Finding the rank(C) = 4. Since the rank of the augmented matrix is not equal to the rank of the original matrix, there is no solution “x”.
