Part 9 : Vector Spaces and Subspaces

R represents a set of all real numbers.

So, R = {1, 1.001, 2.1, 3/2, ……….}.

Tuple and Vector Space

A tuple is an ordered list of numbers.

For example : (1,22,3,21) is a 4-tuple (a tuple with 4 elements).

R² is a set of all real valued 2-tuples, each one of them could be represented as a vector of two components (elements).

Here, e is exponential constant. Its value is approximately 2.718

A space where all the vectors in R² lie is called 2 dimensional real coordinates space or R² vector space.

R² vector space, it consists of all the vectors (that have two elements each) and also their linear combinations

Similarly, R³ is a set of all real valued 3-tuples.

Set of R³

R³ vector space, it contains all the vectors (that have 3 elements each) and their linear combinations

So to generalize this, Vector space Rⁿ consist of all the n-tuple vectors and their linear combinations where each element of tuple is a real number.

Dimensions

Maximum number of linearly independent vectors in a vector space is called Dimension of that vector space. So, if we assume a vector space V, then the dimension of that vector space (dim V) will be maximum number of linearly independent vectors in V.

Basis

Biggest set of linearly independent vectors in a vector space is called Basis.

Or

If one more vector from vector space is added to the Basis set, the set will become linearly dependent.

The number of elements in set of Basis is equal to Dimension of vector space.

Is span a vector space?

Yes, because vector space also the set of all linear combinations of all the vectors inside it just like a span.

If the vectors, A = {a1, a2, a3, …., an} are linearly independent then the basis of vector space would be same as span.

Before we go into vector sub spaces, we will look at Closure law from Discrete Mathematics.

Closure Law

A set is said to be closure under an operation (like addition, subtraction, multiplication, etc.) if that operation is performed on elements of that set and result also lies in set.

For example, N is a set of all natural numbers.

N = {1,2,3,…….}

Now, checking (checking for all the elements would not possible because of enormous number of elements in set) if closure holds for addition for set N.

1 + 2 = 3 (3 lies in set N)

1 + 5 + 100 = 106 (106 lies in set N)

So, closure holds for addition of set N.

Checking for multiplication.

1 x 3 x 6 x 91 = 1638 (lies in set N)

Closure holds for multiplication for set N.

Sub spaces

A nonempty subset of vector space for which closure holds for addition and scalar multiplication is called a subspace.

Read Part 10 : Example of Subspaces

You can view the complete series here
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