Complete the following questions. (a) which two conditions need to be satisfied for a set of vectors in a vector space to form a subspace. (b) according to part (a), determine whether all sequences ū in roo of the form ū = (v,1,0,1,0, form a subspace.

(a) Let be a subset of a vector space . is a subspace of if and only if the following two conditions hold:

i) is closed under the sum operation. That is to say, whenever are elements of .

ii) is closed under the scalar multiplication. That is to say, whenever and

Step-by-step explanation:

For the part (b) we have the set of all sequences of the form, where . Observe the if you multiply any sequence of this form by and scalar then the sequence stops being like the given form. For example, let . Then:

This implies that the set under consideration is not closet under scalar multiplication, which implies that the set is not a subspace of the vector space of all sequences.

is means division by fractions

step-by-step explanation:

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.