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Linear Algebra

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Mathematics
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1. Let V be a vector space over the field R. Two of the properties of the definition of a vector
space are listed below. Prove these properties for R
3
. Prove these properties for P
4.
Explain all
steps.
(i) There exists a 0 vector such that 0 + u = u for all vectors u
V
(vi) (c + d)u = cu + du For all scalars c and d
R and for all vectors u
V
i) Let P4 be the set of all vectors 4x1 and define the vector u such that:
We can always find a 0 vector (with 0 in all the cells) such that:
 
 
 
 
 
vi) Let P4 be the set of all vectors 4x1 and c and d two scalar numbers, and define the vector u
such that:
We can write:
 
 
 
 
 
 
 
 
 
  
 
 
 
 
   
  
2. (a) Let V be a vector space over the field R Let {v
1
, v
2
, v
3
, . . . ,v
n
} be a set of vectors in V. . Define
what it means for this set to span V. Hint start by saying let v
V.
(b) Use the definition of span to prove that the vectors
2
0



and
1
2



span all of R
2
.
(c) State the definition of linearly independence use {v
1
, v
2
, v
3
, . . . ,v
n
})and use it to prove
that the set of vectors in part (b) is linearly independent?
(d) Do the vectors
2
0



and
1
2



form a basis of R
2
?
(e) Determine the coordinates of the vector
3
2



with respect to this basis and indicate this
position on the graph in part (c). Interpret this geometrically
a) Let {u
1
,u
2
} be a linearly independent subset of the set {v
1
,v
2
,v
3
,v
4
} in V and the vectors u
3
and
u
4
be obtained as a linear combination of u
1
and u
2
, according to the spanning set theorem, the
set {u
1
,u
2
} still spans V. In this sense, since they are linearly independent vectors, they are the

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basis for the space V=Span{v
1
,v
2
,v
3
,v
4
}.
b) Taking into account our previous definition, to prove that the vectors
2
0



and
1
2



span R2,
we need to demonstrate that they are linearly independent. We can easily achieve this by
calculating the determinant of the matrix formed by both vectors and proving it is equal to 0.
Hence,

            
Meaning that the two vectors are linearly independent, and therefore span R
2
.
c) It has already been proved that the two vectors are linearly independent because the
determinant of the resulting matrix is equal to 0 (see section b for further details)
d) Yes, considering that they are linearly independent they can be used to define a basis in R
2
e) To establish the coordinates of the vector (3, 2) in the basis formed by the above vectors we
would need to solve the system of equations:
 

Solving this system of equations, we find that the coordinates of the vector (3,2) would be a =
1, b = 1. The graphical representation would be:
As can be observed, the representation of the point (3,2) corresponds to the sum of the vectors (2,0) and
(1,2).

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1. Let V be a vector space over the field R. Two of the properties of the definition of a vector space are listed below. Prove these properties for R3. Prove these properties for P4. Explain all steps. (i) There exists a 0 vector such that 0 + u = u for all vectors u  V (vi) (c + d)u = cu + du For all scalars c and d R and for all vectors u  V i) Let P4 be the set of all vectors 4x1 and define the vector u such that: 𝑎 𝑏 𝑢=( ) 𝑐 𝑑 We can always find a 0 vector (with 0 in all the cells) such that: 𝑎 𝑎+0 𝑏 𝑏+0 𝑢+0=( )=( )=𝑢 𝑐 𝑐+0 𝑑 𝑑+0 vi) Let P4 be the set of all vectors 4x1 and c and d two scalar numbers, and define the vector u such that: 𝑥 𝑦 𝑢=( ) 𝑧 𝑡 We can write: (𝑐 + 𝑑) ∗ 𝑥 𝑥 𝑐∗𝑥 𝑥 𝑥 𝑑∗𝑥 (𝑐 + 𝑑) ∗ 𝑦 𝑦 𝑐∗𝑦 𝑦 𝑦 𝑑∗𝑦 (𝑐 + 𝑑)𝑢 = (𝑐 + 𝑑) ∗ ( ) = ( )=( )+( )=𝑐∗( )+𝑑∗( ) 𝑐∗𝑧 𝑧 𝑧 𝑧 (𝑐 + 𝑑) ∗ 𝑧 𝑑∗𝑧 𝑡 𝑐∗𝑡 𝑡 𝑡 (𝑐 + 𝑑) ∗ 𝑡 𝑑∗𝑡 =𝑐∗𝑢+𝑑∗𝑢 2. (a) Let V be a vector space over the field R Let {v1, v2, v3, . . . ,vn } be a set of vectors in V ...
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