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M303 Linear Algebra: Assignment 2 (1) Determine which of the following are subspace of Rn . If it is a subspace, find a finite set of vectors that span this subspace. \u001a \u0014 \u0015 \u001b x xy = 0 (a) R2 . y x (b) y z (c) \u001a\u0014 \u0015 x y x 2y + 3z = 0 . \u001b x2 y 2 = 0 . (d) Suppose \u0014 1 A= 2 3 1 \u0015 2 . 0 Is the following a subspace of R2 ? \u0014 \u0015 \u001b \u001a\u0014 \u0015 b1 b1 Ax = has a solution . b2 b2 (2) Determine whether 1 1 0 3 span 1 , 1 2 1 1 1 1 (3) Is 1 in the column space of the matrix A = 1 3 1 1 1 . 2 (4) Find a finite set of vectors that span the null space of \u0014 \u0015 1 3 1 3 . 2 1 2 4 (5) Show that given any two subspaces X Rn and Y Rn , the intersection of these subspaces, X Y , is also a subspace of Rn . Note: A vector v is in the intersection, X Y , of subspaces X and Y if v X and v Y . So for instance, if X and Y are subspaces, is 0 in X and in Y ? If so, 0X Y. 1