Let V = R³ and let H be the subset of V of all points on the plane -6x + 7y - 5z = 0. Is H a subspace of the vector space V? 1. Is H nonempty? choose 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>, <4,5,6>. 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, <3,4,5>. 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. choose

Let 4 be a linear combination of {u₁, 1₂, 13). Select the best statement. A. span{u₁, U₂, U3} = span{u₁, U2₂, U3, 14} when u4 is a scalar multiple of one of {U₁, U₂, U3}. B. span{u₁, U₂, U3} = span{U₁, U₂, U3, U4}. c. We only know that span{u₁, U₂, U3} span{u₁, U₂, U3, U4} . D. There is no obvious relationship between span{u₁, U₂, U3} and span{u₁, U₂, U3, U4} . E. none of the above

A King in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of the 64 squares of a chessboard. On the second square, the King would place two grains of​ wheat, on the third​ square, four grains of​ wheat, and on the fourth square eight grains of wheat. If the amount of wheat is doubled in this way on each of the remaining​ squares, how many grains of wheat should be placed on a square 16​? Also, find the total number of grains of wheat on the board at this time and their total weight in pounds.​ (Assume that each grain of wheat weighs​ 1/7000 pounds.)

The set containing the elements specified also possesses its relevant operations of multiplication over reals and vector addition. Is the set a real vector space?