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[answered] Math 240 Homework Assignment Four Do the following problems


I need help with this linear algebra exercise. Please see attached for more detail. Thanks


1.? [6] Show (that is, verify all the axioms) that the set of second degree polynomials

defined as,

????!? =?? ????? ??????? ????? ?????? = ????????! + ???????? + ???? ????????????????? ????, ????, ???? ? ?}

is a vector space. What is the dimension of this vector space?

2.? [4] Consider the set S of 2?3 matrices defined by,

???? =

1??? 0??? 0

0??? 0??? 0

,

0??? 1??? 0

0??? 0??? 0

,

0??? 0??? 0

0??? 1??? 0

,

0??? 0??? 0

0??? 0??? 1

Find the span of S.


3.?? [3] Show that ????!? is a subspace of ????!? (See Problem #1 for the definition of the polynomial

vector spaces.)

4.??? [3] Do the polynomials ????!? + 2???? + 1, ????!? ? ???? + 2, ????!? + 2, and ?????!? + ????!? ? 5???? + 2

span ????!? (Give support for your conclusion.)

5.?? [4] Let S be the set of vectors in ?! (???? =?? ????????, ????????, ????????, ????????, ????????? ) where,

????????? =?? ????, ????, ?????, ????

????????? =?? ?????, ????, ?????, ????

????????? =?? ????, ????, ????, ?????

????????? =?? ?????, ????, ?????, ????

????????? =?? ????, ????, ????, ?????

Find a subset of S that is a basis for the span(S).



Math 240 Homework Assignment Four

 

Do the following problems showing all your work. Be sure to submit your solutions in

 

PDF format to the Assignment area of LEO. (Point values are given in the square

 

brackets.)

 

1. [6] Show (that is, verify all the axioms) that the set of second degree polynomials

 

defined as,

 

! = = ! + + ? , , ? ?}

 

is a vector space. What is the dimension of this vector space?

 

2. [4] Consider the set S of 2?3 matrices defined by,

 

= 1 0 0 0 1 0 0 0 0 0 0 0

 

,

 

,

 

,

 

0 0 0 0 0 0 0 1 0 0 0 1 Find the span of S.

 

3. [3] Show that ! is a subspace of ! (See Problem #1 for the definition of the

 

polynomial vector spaces.)

 

4. [3] Do the polynomials ! + 2 + 1, ! ? + 2, ! + 2, and ? ! + ! ? 5 + 2

 

span ! ? (Give support for your conclusion.) 5. [4] Let S be the set of vectors in ?! ( = , , , , ) where, = = = = = , , ?, ?, , ?, , , , ?

 

?, , ?, , , , ? Find a subset of S that is a basis for the span(S).

 


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