Consider the function . Show that T is a linear transformation. Find the image, kernel, rank, and nullity of T.
the solution is
We know that for any symmetric matrix times , creates a quadratic form(unique), and that the image of T defines the entire space of quadratic form described by the symmetric matrices.
Kernel of T: if A is a skew symmetric matrix, that is then ,
. So, the space of all skew-symmetric matrices is the subset of the kernel of T. By rank-nullity theorem, dimension of kernel of T= nullity of T As a result, because is the dimension of the space of skew-symmetric matrices, all skew-symmetric matrices make up the whole kernel of T.
Kyle is getting some flowers for Olivia, his Valentine. Being of a precise analytical mind, he plans to spend exactly $24 on a bunch of exactly two dozen flowers. At the flower market they have lilies ($3 each), roses ($2 each), and daisies ($0.50 each). Kyle knows that Olivia loves lilies; what is he to do?
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