Study Notes on Vector Space for GATE 2018

Vector Spaces

1-Space (ℜ) = {x | x is a real number};

2-Space (ℜ²) = {(x, y) | x, y are real numbers};

3-Space (ℜ³) = {(x, y, z) | x, y, z are real numbers};

4-Space (ℜ⁴) ={(x₁,x₂,x₃,x₄) | x₁,x₂,x₃,x₄ are real numbers};

n-Space (ℜⁿ) ={(x₁,x₂,…,x­_n) | x₁,x₂,…,x­_n are real numbers}

The elements of are called points or vectors. They are usually denoted by boldface letters as

The ith entry of the vector x=(x₁,x₂,...x_i,…x_n) is called its ith coordinate or its ith component.

The zero vector in ℜⁿ is

0 = (0, 0,…,0).

If x=(x₁,x₂,…x_n) and y=(y₁,y₂,…,y_n) are vectors in ℜⁿ, then their sum is defined as the vector

x + y = (x₁+y₁,x₂+y₂,…,x_n+y_n).

If c is a scalar (a real number), then the scalar multiple of the vector x by the scalar c, denoted by cx, is the vector

cx=(cx₁,cx₂,…,cx_n)

Note:

(-1)x = -x = (-x₁,-x₂,…,-x_n)

Vector Space: Let V be a set vectors in which the operations of sum of vectors and of scalar multiplication are defined (that is, given vectors x and y in V and a scalar c, the vectors x + y and cx are also in V - in this case V is said to be closed under vector addition and multiplication by scalars). Then with these operations V is called a vector space provided that - given any vectors x, y, and z in V and any scalars a and b - the following properties are true:

1. x + y = y + x (commutativity)
2. x + (y + z) = (x + y) + z (associativity)
3. x + 0 = 0 + x = x (zero element)
4. x + (-x) = (-x) + x = 0 (additive inverse)
5. a(x + y) = ax + ay (distributivity)
6. (a + b)x = ax + bx
7. a(bx) = (ab)x
8. (1)x = x

Theorem: The n-space ℜⁿ is a vector space.

Let W be a nonempty subset of the vector space V. If W is a vector space with the operations of addition and scalar multiplication as defined in V, then W is a subspace of V.

Examples:

1. W = {0} is a subspace of ℜⁿ (called the zero subspace).
2. W = ℜⁿ is a subspace of ℜⁿ (also called the improper subspace).
   (all other subspaces of ℜⁿ are called proper subspaces)

Theorem: (Conditions for a subspace)

The nonempty subset W of the vector space V is a subspace of V if and only if it satisfies the following conditions:

1. 0 is in W;
2. If x and y are vectors in W, then x + y is also in W;
3. If x is in W and c is a scalar, then the vector cx is also in W.

Theorem: (Solution subspaces)

If A is an m x n matrix of constants, then the solution set of the homogeneous linear system

Ax = 0

is a subspace of ℜⁿ.

Example: Find two solution vectors u and v for the following homogeneous system such that the solution space is the set of all linear combinations of the form au + bv:

We reduce the coefficient matrix to echelon form by applying the following sequence of EROs:

-3R₁+2R₂, -5R₁+2R₃, -3R₂+R₃

The echelon matrix we obtain is

Hence x and z are the leading variables, and y and w are the free variables. Back substitution yields the general solution

y = a, w = b, z = b, x = -2a + b

Thus the general solution vector of the system has the form

where u = (-2, 1, 0, 0) and v = (1, 0, 1, 1).

The solution space of the system is completely determined by the vectors u and v by the formula x = au + bv.

The vector y is called a linear combination of the vectors x₁,x₂,…,x_n provided that there exists scalars c₁,c₂,…,c_n such that

y=c₁x₁+c₂x₂+…+c_nx_n

Let S ={x₁,x₂,…,x_n} be a set of vectors in the vector space V. The set of all linear combinations of x₁,x₂,…,x_n is called the span of the set S, denoted by span(S) or span (x₁,x₂,…,x_n).

Theorem: span(S) is a subspace of V.

The set S = {x₁,x₂,…,x_n} of vectors in the vector space V is a spanning set for V provided that every vector in V is a linear combination of the vectors in S.