Bob Alexander

07/26/2023 · Senior High School

Question 1 (1.1) The following theorem is called the cancellation law of vector addition: If \( x, y \), and \( z \) are vectors in a vector space \( V \) such that \( x+z=y+z \). Then \( x=y \). (a) Suppose that \( V \) is a vector space over a field \( F \). What do we mean by the zero vector of \( V \) ? (b) Use the cancellation law to prove that the zero vector in \( V \) is unique. (1.2) Let \( V=\mathbb{R}^{2}, \alpha \in \mathbb{R} \) and define: Prove that \( W \) is a subspace of \( V \). (1.3) Use the Lagrange interpolation formula to construct the polynomial of smallest degree whose graph contains the points (1,2), (2,1) and (3,2). Show the details of your calculations.