Is a set of all positive real numbers a vector space?
Recall that the set R+ of all positive real numbers is a vector space with the following definitions for addition and scalar multiplication (to avoid confusion, we use the symbol D to refer to the addition operation, and ∧ to refer to the operation of scalar multiplication): aDb = ab and λ ∧ a = aλ.
Is R+ A vector space over R?
Solution. R+ is not a vector space over R with respect to scalar multiplication defined by λ ⊙ x = λx.
Is R+ A field?
Since we found one vector which spans, R+ is a one-dimensional vector space over the field R. If we use a subfield of R for F, then depending on which x we choose, log2 x may or may not be an element of F, and so we will need more than one element in our basis for this other vector space.
Is the set of all integers a vector space?
(a) The set of all integers. This set will not form a vector space because it is not closed under scalar multiplication. When, the scalar, which can take any value, is multiplied by the integer, the resulting number may be a real number or rational number or irrational number or integer.
Is a vector any element of a vector space?
A vector is any element of a vector space. A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u+v are in H, and (iii) c is a scalar and cu is in H.
Can inner product negative?
The inner product is negative semidefinite, or simply negative, if ‖x‖2≤0 always. The inner product is negative definite if it is both positive and definite, in other words if ‖x‖2<0 whenever x≠0.
Is C an R vector space?
(i) Yes, C is a vector space over R. Since every complex number is uniquely expressible in the form a + bi with a, b ∈ R we see that (1, i) is a basis for C over R. Thus the dimension is two. (ii) Every field is always a 1-dimensional vector space over itself.
Is Za a field?
The integers (Z,+,×) do not form a field.
Which one is not a vector space?
Similarily, a vector space needs to allow any scalar multiplication, including negative scalings, so the first quadrant of the plane (even including the coordinate axes and the origin) is not a vector space.
Which set is not a vector space?
1 Non-Examples. The solution set to a linear non-homogeneous equation is not a vector space because it does not contain the zero vector and therefore fails (iv). is {(10)+c(−11)|c∈ℜ}. The vector (00) is not in this set.
Which is a vector space in its own right?
That plane is a vector space in its own right. If we add two vectors in the plane, their sum is in the plane. If we multiply an in-plane vector by2or 5, it is still in the plane. A plane in three-dimensional space is notR2 (even if it looks like R2/.
How much thrust does a Space Shuttle have?
Photo credit: NASA After the solid rockets are jettisoned, the main engines provide thrust which accelerates the Shuttle from 4,828 kilometers per hour (3,000 mph) to over 27,358 kilometers per hour (17,000 mph) in just six minutes to reach orbit. They create a combined maximum thrust of more than 1.2 million pounds.
How does the main engine on a space shuttle work?
The main engines develop thrust by using high-energy propellants in a staged combustion cycle. The propellants are partially combusted in dual preburners to produce high-pressure hot gas to drive the turbopumps. Combustion is completed in the main combustion chamber.
How is the vector space R2 represented in math?
The vector space R2 is represented by the usual xy plane. Each vector v in R2 has two components. The word “space” asks us to think of all those vectors—the whole plane. Each vector gives the x and y coordinates of a point in the plane: v D.x;y/. Similarly the vectors in R3 correspond to points .x;y;z/ in three-dimensional space.