Question

1 point) Let V R2 and let H be the subset of V of all points on the line-4x-3y-0. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? | H does not contain the zero vector of V | 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as <1,2>, <3,4> 2,2>, <3,2> 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and syntax such as 2, <3,4>. 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. H is not a subspace of V

Answer 1

The sets {0}, R² and any set L of the form H = {kv : k∈ R, v≠ 0} which consists of all scalar multiples of a non-zero vector v are the only subspaces of R². The line -4x-3y = 0 contains the point (0,0) so that it is a subspace of R².

1. YES. The line H contains the origin as -4*0-3*0 = 0.

2. YES. The line H is closed under vector addition. If X = (x,y) and Y = (p,q) are 2 vectors such that = -4x-3y = 0 and -4p-3q = 0, then -4(x+p)-3(y+q) = -4x-3y-4p-3q = 0+0= 0.

3. YES. The line H is closed under scalar multiplication. If -4x-3y = 0, then -4kx-4ky = k(-4x-3y) = k*0 = 0.

4. Since H contains the origin and is closed under vector addition and scalar multiplication, hence H is a vector space and, therefore, a subspace of R².