Determine whether the set of vectors is a basis for ℛ3. Given the set of vectors , decide which of the following statements is true: A: Set is linearly independent and spans ℛ 3. Set is a basis for ℛ 3. B: Set is linearly independent but does not span ℛ 3. Set is not a basis for ℛ 3. C: Set spans ℛ 3 but is not linearly independent. Set is not a basis for ℛ 3. D: Set is not linearly independent and does not span ℛ 3. Set is not a basis for ℛ 3.

(A) Set A is linearly independent and spans . Set is a basis for .

Step-by-Step Explanation

Definition (Linear Independence)

A set of vectors is said to be linearly independent if at least one of the vectors can be written as a linear combination of the others. The identity matrix is linearly independent.

Definition (Span of a Set of Vectors)

The Span of a set of vectors is the set of all linear combinations of the vectors.

Definition (A Basis of a Subspace).

A subset B of a vector space V is called a basis if: (1)B is linearly independent, and; (2) B is a spanning set of V.

Given the set of vectors   , we are to decide which of the given statements is true:

In Matrix , the circled numbers are the pivots. There are 3 pivots in this case. By the theorem that The Row Rank=Column Rank of a Matrix, the column rank of A is 3. Thus there are 3 linearly independent columns of A and one linearly dependent column. has a dimension of 3, thus any 3 linearly independent vectors will span it. We conclude thus that the columns of A spans .

Therefore Set A is linearly independent and spans . Thus it is basis for .

the answer   is 200

step-by-step explanation:

this is a problem about poisson probabilities.   i don't know if you've

already studied poisson distributions, or if you have a place to look

to read more about the subject.  

here's how you can see a poisson distribution coming.   when there are

a lot of events that may happen at any time (or any place), and when

you're counting how many of them happen at a particular time (or a

particular place), and each event is independent of the others, then

the result is a poisson distribution.   this is roughly true for the

chocolate chips; one chip being in a cookie has only a small effect on

the probability that another chip will be in the cookie.

once you know you have a poisson distribution, you know a lot.   there

is only one parameter to a poisson distribution, so if you know the

mean, you know everything.   the form of the distribution is

  probability of n chips = {[e^(-x)]*(x^n)} / (n! )

your answer would be 12