Question
In: Advanced Math
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not identified. V = R^2 , < X1 , X2 > + < Y1 , Y2 > = < X1 + Y1 , 0> c< X1 , X2 > = < cX1 , cX2 >
Solutions
Colby Messinger answered 2 years agoRelated Solutions
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = { (x1, x2, x3) ∈ R^3 | x1 > or equal to 0,
x2 > or equal to 0, x3 > or equal to 0}
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = { f : R --> R | f(1) = 0 }
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = {all polynomials with real coefficients with
degree > or equal to 3 and the zero polynomial}
The set R^2 with addition and scalar multiplication defined by (x1, y1) + (x2, y2) =...
The set R^2 with addition and scalar multiplication defined
by
(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
c(x1, y1) = (cx1, y1)
is not a vector space. Determine which axiom fails and find a
counterexample that shows that it fails.
Let ?V be the set of vectors in ?2R2 with the following definition of addition and...
Let ?V be the set of vectors in ?2R2 with the following
definition of addition and scalar multiplication:
Addition: [?1?2]⊕[?1?2]=[0?2+?2][x1x2]⊕[y1y2]=[0x2+y2]
Scalar Multiplication: ?⊙[?1?2]=[0??2]α⊙[x1x2]=[0αx2]
Determine which of the Vector Space Axioms are satisfied.
A1. ?⊕?=?⊕?x⊕y=y⊕x for any ?x and ?y in ?V
? YES NO
A2. (?⊕?)⊕?=?⊕(?⊕?)(x⊕y)⊕z=x⊕(y⊕z) for any ?,?x,y and ?z in
?V
? YES NO
A3. There exists an element 00 in ?V such that ?⊕0=?x⊕0=x for
each ?∈?x∈V
? YES NO
A4. For each ?∈?x∈V, there exists an...
Determine whether the members of the given set of vectors are linearly independent. If they are...
Determine whether the members of the given set of vectors are
linearly independent. If they are linearly dependent, find a linear
relation among them of the form c1x(1) + c2x(2) + c3x(3) = 0. (Give
c1, c2, and c3 as real numbers. If the vectors are linearly
independent, enter INDEPENDENT.) x(1) = 9 1 0 , x(2) = 0 1 0 , x(3)
= −1 9 0
Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and...
Let S be the set of all ordered pairs of real numbers. Define
scalar multiplication and addition on S by:
α(x1,x2)=(αx1,αx2)
(x1,x2)⊕(y1,y2)=(x1 +y1,0)
We use the symbol⊕to denote the addition operation for this
system in order to avoid confusion with the usual addition x+y of
row vectors.
Show that S, together with the ordinary scalar multiplication
and the addition operation⊕, is not a vector space.
Test ALL of the eight axioms and report which axioms
fail to hold.
Determine whether each set of vectors is linearly dependent or linearly independent. a) (1,1,0,1), (1,0,1,1), (0,1,1,1)...
Determine whether each set of vectors is linearly dependent or
linearly independent.
a) (1,1,0,1), (1,0,1,1), (0,1,1,1)
b) (1,0,1,0), (0,1,0,1), (1,-1,1,-1), (1,-1,0,0)
Set up the addition and multiplication tables for Z3 and Z6. Use these tables to verify...
Set up the addition and multiplication tables for Z3 and
Z6.
Use these tables to verify that (Z3, +), (Z3 \ {0}, ·) and (Z6, +)
are groups, but (Z6 \ {0}, ·) is not a group.
In which finite field does "25 divided by 5 is 14" hold?
Vectors in the plan. Define scalar product and explain the relationship between scalar product defined by...
Vectors in the plan.
Define scalar product and explain the relationship between
scalar product defined by coordinates respectively at lengths and
angles between vectors.
Significance of the scalar product sign.
The determinant of a vector pair.
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