Vector Space Assessments Paper

This paper discusses questions 3.7.24-3.7.26 on vector space assessments, including spectrum of alpha, nilpotent operators, and inner products. It also covers the orthogonality and positivity of quantities in real and complex vectors. Course code, name, and college/university are not mentioned.

Running head: VECTOR SPACE ASSESSMENTS
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Vector Space Assessments Paper
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Vector Space Assessments
Question 3.7.24
Part 1
Since the spectrum of α is for values of μi where i assumes all positive values.
∝=ui
S ( ∝ )=ui
Sk ( ∝ ) =ui 1 , … ,uik
For n dimensional vector space we will assume value i=1,…,∞ . Hence, 1 ≤ii , … ,i∞ ≥ n
SK (∝)={μi 1 , … , μik } for n dimensions i=1,…,K . Hence, 1 ≤ii , … ,iK ≥ n (Rotman, 2003, p. 172)
Hence
SK ( ∝ )= { μi 1 ,… , μik }1 ≤ ii , … , iK ≥ n
Part 2
Since the spectrum of α is for values of μi where i1<i2
∝=ui
∧ ( ∝ )=ui
∧k ( ∝ ) =ui 1 , … , uik
For n dimensional vector space we will assume value i=1,…,∞ . Hence, 1 ≤ii <…<i∞ ≥ n

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(Rotman, 2003, p. 172).
∧K (∝)={ μi 1 , … , μik } for n dimensions which will assume value 1 to N for i=1<…<K. Hence,
1 ≤ii <…<iK ≥n
Hence
∧K (∝)= {μi 1 , … , μik }1 ≤ii <…<iK ≥ n
Question 3.7.25
If we assume the following dim V =n≥ 1
Then we can suppose that ∝ is a nilpotent operator i.e. ∝q=0
We can therefore consider that {0}⊆ ∧ ( ∝ ) ⊆∧ ( ∝2 ) ⊆ ∧ ( ∝q )=V
If x ∈ ∧ ( ∝q ) then ∝q ( x ) =0 so ∝q+ 1 ( x )=∝ ( ∝q ( x ) )=∝ ( 0 )=0
Therefore
if q ≥ 1 then∝=0
As such
tr (∧¿¿ q ( ∝ ))=tr (∧q ( 0 ) )=0 ¿ CITATION Jos03 \p 383-384 \l 1033 (Rotman, 2003, pp. 383-384)

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Question 3.7.26
Since V and U have values that are real numbers then their inner products will only assume
combinations of this real numbers Regardless of how many times the values are repeated. Real
numbers assume values between -∞ and ∞ (Roman, 2005). Hence, we can simply this by writing
Quantities (u,´u) and (v,´v) are always real even if V and U are complex vectors (Vinberg, 2003)
¿ . , .>:V ⊗ U ≈< . ,.>: ( V ⊗U ) ∗(U ⊗ V )
λ λ ( u ,u' ) = ( λu , λ u' )= ( U ( u ) , U ( u' ) ) =(u ,u' )
λ λ ( u ,u' ) = ( λu , λ u' )= ( V ( v ) ,V ( v' ) )=(v , v' )
They are orthogonal (Axler, 2015)
( V ( v ) , V ( v' ) ) = ( v , v' ) =δvv '
(U ( u ) , U ( u' ) )= (u , u' )=δuu '
With regard to positivity
( v , v' ) ≥ 0 for all v , v ' ∈V
( u , u' ) ≥0 for all u , u ' ∈U
Hence
¿ . , .>:V ⊗ U → R thenu , v ,u' , v' ∈ V ×U

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References
Axler, S. (2015). Linear Algebra Done Right. Springer: New York.
Hoffman, K. M., & Kunze, . R. (1971). Linear Algebra (2nd Edition). Pearson: London.
Loehr, N. (2014). Advanced Linear Algebra. Florida: Chapman and Hall Publishers/CRC.
Roman, S. (2008). Advanced Linear Algebra. New York: Springer.
Rotman, J. J. (2003). Advanced Modern Algebra. Upper Saddle River: Prentice Hall.
Vinberg, E. B. (2003). A Course in Algebra . Providence: American Mathematical Society.

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