Consider the mapping H: R² – R* where H(x1, X2)=(X1, X2, 2x1, X1+X2). Prove that H is a linear transformation. (This is a projection into higher dimension. Vladimir Vapnik successfully used similar technique to develop support vector machine, as his significant contribution to machine learning.)

The cross product of two vectors in R³ is defined by Let = A = a₁ a2 a3 b1 [a2b3-a3b2] b2 = a3b1a1b3 ხვ La1b2-a2b1 H 1 Find the matrix A of the linear transformation from R³ to R³ given by T(z) = x *.

Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D : P3 - P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) = P3. Is D a linear transformation? Let p(x) = a³x³ + ²x² + a₁ + a₁ and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆

(1) Prove that any linear transformation T : R" → R" maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?