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If a is a vector:

sage: a = vector(QQ, [1,2,3,4]) ; a
(1, 2, 3, 4)

The scalar product leads to:

sage: 5*a
(5, 10, 15, 20)

And the sum is:

sage: a+a
(2, 4, 6, 8)

For lists; the sum works as you expect (catenation):

sage: [1,2,3] + [4,5,6]
[1, 2, 3, 4, 5, 6]

You can easily transform a vector into a list:

sage: list(a)
[1, 2, 3, 4]

To transform a list into a vector, it is safer to provide the base ring:

sage: vector(QQ,[1,2,3,4])
(1, 2, 3, 4)
sage: vector(RDF,[1,2,3,4])
(1.0, 2.0, 3.0, 4.0)

sage: vector(ZZ,[1,2,3,4]).parent()
Ambient free module of rank 4 over the principal ideal domain Integer Ring
sage: vector(QQ,[1,2,3,4]).parent()
Vector space of dimension 4 over Rational Field

So, to sum up, you can transform your vectors into litst, sum them and make them vectors again, by keeping the base_ring information:

sage: vector(a.base_ring(),list(a)+list(5*a))
(1, 2, 3, 4, 5, 10, 15, 20)

If a is a vector:

sage: a = vector(QQ, [1,2,3,4]) ; a
(1, 2, 3, 4)

The scalar product leads to:

sage: 5*a
(5, 10, 15, 20)

And the sum is:

sage: a+a
(2, 4, 6, 8)

For lists; the sum works as you expect (catenation):

sage: [1,2,3] + [4,5,6]
[1, 2, 3, 4, 5, 6]

You can easily transform a vector into a list:

sage: list(a)
[1, 2, 3, 4]

To transform a list into a vector, it is safer to provide the base ring:

sage: vector(QQ,[1,2,3,4])
(1, 2, 3, 4)
sage: vector(RDF,[1,2,3,4])
(1.0, 2.0, 3.0, 4.0)

sage: vector(ZZ,[1,2,3,4]).parent()
Ambient free module of rank 4 over the principal ideal domain Integer Ring
sage: vector(QQ,[1,2,3,4]).parent()
Vector space of dimension 4 over Rational Field

So, to sum up, you can transform your vectors into litst, sum them and make them vectors again, by keeping the base_ring information:

sage: vector(a.base_ring(),list(a)+list(5*a))
(1, 2, 3, 4, 5, 10, 15, 20)

sage: A = matrix(QQ, 2, 2, [3,9,6,10]) ; A
[ 3  9]
[ 6 10]
sage: block_matrix([[A, -A], [10*A, 100*A]], subdivide=False)
[   3    9   -3   -9]
[   6   10   -6  -10]
[  30   90  300  900]
[  60  100  600 1000]