31. :Note that the important part here is the " p " linearly independent eigenvectors. 32. Fourth, a matrix is invertible if the columns are linearly independent , hence P is invertible. 33. Since this is a second-order differential equation, we must have two linearly independent solutions. 34. The strain rate tensor is symmetric by definition, so it has only six linearly independent elements. 35. The atoms in A as well as the atoms in B are assumed to be linearly independent . 36. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be geometric multiplicity. 37. One seeks conditions on the existence of a collection of solutions such that the gradients are linearly independent . 38. Of which the first three are easily seen to be linearly independent , and therefore span all of. 39. The most general solution will be a linear combination of linearly independent solutions to the above homogeneous equation. 40. I now have to find a linearly independent branch, which I am unsure of how to do.
