# LINEAR ALGEBRA Competency 209.8.4: Vector Spaces

August 30, 2017

Question
Upon further inquiry I received this more specific feedback:

W is a subspace where the x-coordinate of the vector is equal to the y-coordinate of the vector. Use this argument to show that vector (0,0) is in the subspace W.

Next provide concrete examples of other vectors that are in W. For example, you might state that (1,1) are (2,2) are within W. This proves that W is not trivial. That is, W is not a singleton.

Finally, provide a few examples of vectors that are not in W. For instance, the vector (3,5) is not in W. Why? Explain this to illustrate that W is a PROPER subspace of the 2D vector space of real numbers. This will show that W is not all of the vector space that contains it.

LINEAR ALGEBRA
Competency 209.8.4: Vector Spaces – The graduate demonstrates understanding
of the properties and characteristics of vector spaces.
Introduction:
Vector spaces possess a collection of specific characteristics and properties. Use the
definitions in the attached “Definitions” to complete this task.
Define the elements belonging to R2 as {(a, b) | a, b ∈ R}. Combining elements
within this set under the operations of vector addition and scalar multiplication
should use the following notation:
Vector Addition Example: (–2, 10) + (–5, 0) = (–2 – 5, 10 + 0) = (–7, 10)
Scalar Multiplication Example: –10 × (1, –7) = (–10 × 1, –10 × –7) = (–10, 70), where
–10 is a scalar.
Under these definitions for the operations, it can be rigorously proven that R2 is a
vector space.
Given:
Laws 1 and 2 in the attached “Definitions” are true.
Requirements:
Provide a written explanation (suggested length of 2–4 pages) of why R2 is a vector
space in which you do the following:
A. Prove the truth of Laws 3 through 10 of the provided mathematical definition for a
vector space.
B. Give an example of a subset of R2 that is a nontrivial subspace of R2, showing all
work.
C. When you use sources, include all in-text citations and references in APA format.
Note: For definitions of terms commonly used in the rubric, see the Rubric Terms web
link included in the Evaluation Procedures section.
Note: When using sources to support ideas and elements in a paper or project, the
submission MUST include APA formatted in-text citations with a corresponding
reference list for any direct quotes or paraphrasing. It is not necessary to list sources
that were consulted if they have not been quoted or paraphrased in the text of the
paper or project.
Note: No more than a combined total of 30% of a submission can be directly quoted
or closely paraphrased from sources, even if cited correctly. For tips on using APA
style, please refer to the APA Handout web link included in the General Instructions
section.

The Definition of a Vector Space:
A vector space is a set V of elements (called vectors) together with two operations,
vector addition and scalar multiplication, satisfying the following 10 laws (for all
vectors X, Y, and Z in V and all (real) scalars r and s).
If X and Y are any two vectors in V, then X + Y ∈ V.
2. Associative law
(X + Y) + Z = X + (Y + Z)
3. Commutative law
X+Y=Y+X
There is a vector in V, denote 0 such that X + 0 = X where 0 is called the
zero vector.
For every X ∈ V there is a vector –X such that X + (–X) = 0, where –X is
called the additive inverse of X.
Laws for Scalar Multiplication
6. Closure under scalar multiplication
If X is any vector in V and r is any real scalar, then rX ∈ V.
7. Associative law
(rs)X = r(sX)
8. Distributivity over vector sums law
r(X + Y) = rX + rY
9. Distributivity over scalar sums law
(r + s)X = rX + sX
10. Scalar multiplication identity law
1X = X

The Definition of a Nontrivial Subspace of a Vector Space:
Any subspace W of V that is neither V nor {0} is a nontrivial subspace (note that the
set W needs to be proven a subspace before it can be proven to be a nontrivial
subspace).

B. Give an example of a subset of R2 that is a nontrivial subspace of R2, showing all work.

Consider a line y=x on the plane R2
Define W = {( x , y ) ∈ R2| y=x }
Clearly W is a subset of the vector space V = R2
Now to prove that W is a vector space, by showing all conditions with respect to operation on vector
space.
Let ( x 1 , x 1 ) ∈ W ∧( x2 , x2 ) ∈W ,then ( x1 , x1 ) + ( x2 , x2 ) =( x 1+ x 2 , x 1 + x 2 )
Since x 1+ x2=x 1 + x 2
¿> ( x 1 + x 2 , x 1+ x 2 ) ∈W thus closure property holds .
Let ( x 1 , x 1 ) ∈ W , ( x 2 , x 2) ∈ W ∧( x 3 , x 3 ) ∈W then ,
{( x 1 , x 1 ) + ( x 2 , x 2 ) }+ ( x3 , x3 ) =( x 1+ x2 , x1 + x 2 ) + ( x 3 , x 3 )
¿ ( x 1+x 2 + x 3 , x 1+ x2 +x 3 )
¿ ( x 1 , x 1 )+ ( x 2 + x 3 , x 2 + x 3 )
¿ ( x 1 , x 1 )+ { ( x 2 , x 2 ) + ( x 3 , x 3 ) }
Thus associative law satisfied .
Let ( x 1 , x 1 ) ∈ W , ( x 2 , x 2) ∈ W then
( x 1 , x 1 ) + ( x 2 , x 2 )=( x 1 + x 2 , x 1+ x 2 )
¿ ( x 2+ x 1 , x 2 + x 1 )
¿ ( x2 , x2 )+( x1 , x1 )
Thus commutative law satisfied .

Choose ( 0,0 ) ∈W then for every ( x , x ) ∈V such that
( x , x ) + ( 0,0 )=( x +0, x+ 0 )
¿(x, x)
¿ ( 0+ x ,0+ x )
¿ ( 0,0 ) + ( x , x )
¿(x, x)
¿> ( x , x ) + ( 0,0 )=( x , x )=( 0,0 ) + ( x , x ) .
Hence(0,0)iscalled identity element of W ( ¿ zero vector ) with respect ¿ adddition .
For each ( x , x ) ∈W there is an element (−x ,−x) ∈W such that
( x , x ) + (−x ,−x )= ( x−x , x−x )
¿ ( 0,0 )
¿ (−x + x ,−x +x )

¿ (−x ,−x ) + ( x , x )
¿ ( x , x )+ (−x ,−x )
¿ ( 0,0 )
¿ (−x ,−x ) + ( x , x ) .
¿> (−x ,−x ) is inverse for ( x , x ) with respect ¿ addition .

Scaler Multiplication:
6.) Closure under scaler multiplication:
( x , x ) ∈ W ∧for any a ∈ R then a ( x , x )=( ax , ax ) ∈W
Since product of two real number is real number .
7.) Associative Law:
Let a , b ∈ R∧( x , x ) ∈ W the n
a ( b ( x , x ) ) =a ( bx ,bx )
¿ ( abx , abx )
¿ ( ab )( x , x )

8.) Distributivity of Scaler Multiplication with respect to vector addition:
Let a ∈ R∧( x 1 , x 1 ) ∈ W , ( x 2 , x 2) ∈ W then,
a {( x 1 , x 1 ) + ( x 2 , x 2 ) }=a ( x 1 + x 2 , x 1+ x 2 )
¿ ( a { x 1 + x 2 }+ a { x 1+ x 2 } )
¿ ( a x 1+a x2 , a x 1+ a x2 )
¿ ( a x1 , a x1 )+(a x2 , a x2 )
¿ a ( x 1 , x 1 ) +a ( x 2 , x 2 ) .
9.) Distributivity of Scaler Multiplication with respect to field addition:
Let a , b ∈ R∧( x , x ) ∈ W then
( a+b )( x , x ) =( ( a+b ) x , ( a+b ) x )
¿ ( ax +bx , ax+bx )
¿ ( ax , ax ) + ( bx , bx )
¿ a ( x , x ) +b ( x , x )
10.) Scaler Multiplicative Identity Law:
Let 1∈ R∧any ( x , x ) ∈W then ,
1 ( x , x ) =( 1 x ,1 x )
¿ ( x , x ).

Thus, W satisfies all the condition of vector subspace, therefore, W is vector subspace.

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