
Matrices, Vector Spaces and Zero Vector
To prove that V is a vector space, we need to prove that V is closed under finite vector addition and scalar multiplication.

vector addition velocities
The problem is an illustration of vector addition
Answer
The problem is an illustration of vector addition
Velocity of the plane vplane =156 m/s (it is a vector along east)
Velocity of the wind vwind = 20 m/s ( It is a vector along south )

Matrices, Vector Spaces and Subspaces
as
c) all 2 X 2 matrices A such that det(A) = 0
It is NOT a vector space, since V is NOT closed under finite vector addition.

A vector space V
Definition: A vector space V is a set of elements together with two operations, addition and scalar multiplication, satisfying the following properties:
Let u,v, and w be vectors in V, and let c and d be scalars.
Addition:
(a) u+v is in V.

The null and inverse vectors
Addition is defined as:
(a,b,c)+(d,e,f) = (a+d,b+e, c+f)
Scalar multiplication is defined as:
x(a,b,c) = (xa,xb,xc)
Write down the null vector and inverse of (a,b,c).
show that the vectors (a,b,c) do not form a vector space.

Vector Space and Subspace
Subspace
A nonempty subset S of a vector space V is called a subspace of V, if S is also a vector space over using the same vector addition and scalar multiplication operation.

Vector spaces
Combining elements within this set under the operations of addition and scalar multiplication should use the following notation:
Addition Example: (2, 10) + (5, 0) = (2  5, 10 + 0) = (7, 10)
Scalar Multiplication Example: 10 × (1, 7)

Linear Algebra: Vector Spaces
The unique vector associated with it is:
[a(n1), a(n2),....a(1),a(0)]
Simply define addition coordinatewise as you did for the vector space in R2, and scalar multiplication similarly.

Let F be the field of real numbers and let V be the set of all sequences ( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise.
Prove that V is a vector space over F.
. , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise.
Prove that V is a vector space over F.

Linear Algebra And Differential Equations: Real Vector Space
is an addition '+' in V such that V,+ is a commutative group.