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Linear Algebra Solved Problems with DETAILED explanation in Tamil

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Linear Algebra - Null space ... explanation

Linear algebra question

Which of the followings are vector spaces under usual operations 1.Q over Q
2.Q over R
3.Q over C
4.R over Q
5.C over Q
6.R over R
7.R over C
8.C over R
9.C over C.
Find basis & dimension for each vector space.
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Algebra fact

Algebra fact: Rotation can be linear or nonlinear depending on what you think is fixed. "Rotate v by angle t" is linear in v, not in t. - John D cook

linear algebra

Vector Space 

Definition  

Suppose that

V is a set upon which we have defined two operations: (1) vector addition, which combines two elements of V and is denoted by “+”, and (2) scalar multiplication, which combines a complex number with an element of V.

 . Then V, along with the two operations, is a vector space over C if the following ten properties hold.

• AC Additive Closure
  If u,v∈V, then u+v∈V.
• SC Scalar Closure
  If α∈C and u∈V, then αu∈V.
• C Commutativity
  If u,v∈V, then u+v=v+u.
• AA Additive Associativity
  If u,v,w∈V, then u+(v+w)=(u+v)+w.
• Z Zero Vector
  There is a vector, 0, called the zero vector, such that u+0=u for all u∈V.
• AI Additive Inverses
  If u∈V, then there exists a vector −u∈V so that u+(−u)=0.
• SMA Scalar Multiplication Associativity
  If α,β∈C and u∈V, then α(βu)=(αβ)u.
• DVA Distributivity across Vector Addition
  If α∈C and u,v∈V, then α(u+v)=αu+αv.
• DSA Distributivity across Scalar Addition
  If α,β∈C and u∈V, then (α+β)u=αu+βu.
• O One
  If u∈V, then 1u=u.

The objects in V are called vectors, no matter what else they might really be, simply by virtue of being elements of a vector space.