[Linear_Algebra] Vector space

In this post, we are going to talk about the Vector space.

 

 

Vector space

A vector space is a set of vectors. A vector space must satisfy 3 conditions as follows:

Let x, y be elements in V.

1. x + y is in V (closed under addition)
2. kx is in V for any scalar k (closed under scalar multiplication)
3. the origin vector (0) is in V

Conclusion: if k1x + k2y is in V for any scalars k1, k2, then we call V a vector space.

Example: R3 is a vector space of vectors [x, y, z]. The set of all m×n matrices, written as A^m×n, is also a vector space.

 

 

Subspace

A subspace is a subset of a vector space that satisfies the same 3 conditions.

Example: In R3,

1. A plane passing through the origin (ax + by + cz = 0) is a subspace of R3.
2. A line passing through the origin (ax + by = 0) is a subspace of R3.

 

 

Column Space of Matrix 

Let A; A consists of column vectors.

So if we think about the equation: Ax = b ; where x = [x1, x2, ..., xn].

Ax = b → a1x1 + a2x2 + ... + anxn = b.

This form is a linear combination of the column vectors of A. If the result equals b, then we say that b is in the Column space of A (C(A)).

Example: A = [[1,5,2],[0,4,4]]. The column space of A is a plane in R^2.

Example: If A is a non-singular (invertible) n×n matrix, then C(A) = R^n.

Another example: A = [[100],[010],[110]]. When Gaussian elimination is applied, this matrix has two pivots. So the column space of A is R^2.

Conclusion: In Gaussian elimination of A, the number of pivots equals the dimension of C(A).