10 points Tensor Products Given two vectors v and w, we can form their tensor product or outer product vw, read "v tensor w". The tensor product of two n x 1 vectors is an n x n matrix. The formula for the tensor product is: vw = vw Tensor products are useful for understanding fluids and strength of materials. (a) 4 points Suppose that e, and e₂ are the following vectors in R²: - 8 e₁ = e2= - 8 These vectors are the standard unit vectors for the x and y axes. Calculate all four possible tensor products: e₁ e₁, e₁ ez, ez e₁, and e₂ e2. (b) 2 points If v = [ry], calculate (e₁ & e₁)v and show that your result is the component of v in the direction of e₁. (The fancy way to say this is "the projection of v onto the span of e₁". Tensor products are a convenient way to calculate projection operators in general, especially in different coordinate systems.) (c) 2 points If you wanted to find the component of v in the direction of e2, which tensor product could you use? (d) 2 points Calculate e₁ e₁ + e2e2. What special matrix is this?