The nullspace of a matrix aa is the solution space of the homogeneous system ax=0ax=0. b. a linear system is solvable if and only if the vector of constants belongs to the column space of the coefficient matrix of the system. c. a subspace of a vector space vv cannot contain only a finite number of elements. d. the span of the columns of a matrix is called the column space of the matrix. e. a nonempty subset ww of some vector space vv is called a subspace of vv if ww is closed under linear combinations. f. the set of all vectors of the form [1,x, y]t[1,x, y]t, where xx and yy range over all real numbers, is a subspace of r3r3. g. the set of all vectors of the form [0,x, y]t[0,x, y]t, where xx and yy range over all real numbers, is a subspace of r3r3. h. none of the above

Which graph represents the following piecewise defined function?

The ratios 3: 9, 4: 12, 5: 15, and 6: 18 are a pattern of equivalent ratios in the ratio table. which is another ratio in this pattern?

Cone w has a radius of 8 cm and a height of 5 cm. square pyramid x has the same base area and height as cone w. paul and manuel disagree on how the volumes of cone w and square pyramid x are related. examine their arguments. which statement explains whose argument is correct and why? paul manuel the volume of square pyramid x is equal to the volume of cone w. this can be proven by finding the base area and volume of cone w, along with the volume of square pyramid x. the base area of cone w is π(r2) = π(82) = 200.96 cm2. the volume of cone w is one third(area of base)(h) = one third third(200.96)(5) = 334.93 cm3. the volume of square pyramid x is one third(area of base)(h) = one third(200.96)(5) = 334.93 cm3. the volume of square pyramid x is three times the volume of cone w. this can be proven by finding the base area and volume of cone w, along with the volume of square pyramid x. the base area of cone w is π(r2) = π(82) = 200.96 cm2. the volume of cone w is one third(area of base)(h) = one third(200.96)(5) = 334.93 cm3. the volume of square pyramid x is (area of base)(h) = (200.96)(5) = 1,004.8 cm3. paul's argument is correct; manuel used the incorrect formula to find the volume of square pyramid x. paul's argument is correct; manuel used the incorrect base area to find the volume of square pyramid x. manuel's argument is correct; paul used the incorrect formula to find the volume of square pyramid x. manuel's argument is correct; paul used the incorrect base area to find the volume of square pyramid x.