Part 10 : Example of Subspaces

Read Part 9 : Vector Spaces and Subspaces to get clarity on Rⁿ vector spaces and Closure Law.

Assuming that we have a vector space R³, it contains all the real valued 3-tuples that could be represented as vectors (vectors with 3 real number components).

Set R³

So a subspace of vector space R³ will be a set of vectors that have closure under addition and scalar multiplication.

Taking a subset of R³ with three vectors a, b and c.

V is a subset of R³

V will be a subspace only when :

1. a, b and c have closure under addition i.e. a+b+c, a+b, b+c, etc. should lie in set V.
2. a, b and c have closure under scalar multiplication i.e. 3a, 2b, 0c, etc. should lie in set V.

Zero Vector

A zero vector has zeros as components (number of components depends on its vector space). It is also known as null vector.

For R³ vector space, zero vector will be

Vector space is R³ so zero vector has 3 components

Interestingly, this zero vector is also a subspace of R³ vector space.

Proof

1. Sum of components of zero vector will always be zero. Hence, zero vector holds closure under addition.
2. Scalar multiplication of zero will result in zero vector itself. It holds closure under scalar multiplication.

Thus, zero vector is a subspace of every vector space. It is also called trivial subspace.

Line and Plane passing through Zero Vector

Assuming a line in R³ vector space passing through zero vector, that expands in all 3 dimensions till infinity.

Line Passing through Zero Vector in R³ vector space

It holds closure under addition, because if we add any vectors that lie on that line their sum will also be on that line.

It also holds closure under scalar multiplication, because multiplying any scalar with a vector on that line will result in another vector on that same line (as line extends till infinity in all dimensions).

So, every line that going through zero vector of vector space is a subspace.

Now, assuming a plane through zero vector in R³ vector space (that expands till infinity in all dimensions).

Plane through Zero Vector in R³ vector space

This plane is also a subspace of R³ vector space.

Proof

1. Adding any vectors from that plane will result in a new vector that’s also on that plane.
2. Multiplication of scalar with any vector of that plane will result in another vector that lies in that plane.

Can a vector space be a subspace?

Yes

A set is also a subset of itself. So, R³ is a subset of itself and it also holds closure under addition and scalar multiplication (because it has all the vectors will 3 components).

To Summarize

1. A subspace is a subset of vector space that holds closure under addition and scalar multiplication.
2. Zero vector is a subspace of every vector space.
3. Vector space is a subspace of itself.
4. All the geometric figures having dimension less than the dimension of vector space and passing through zero vector of vector space.

Example :

For vector space R⁴ (4 Dimensional), subspaces are :

a. R⁴ itself

b. Zero vector ([0,0,0,0])

c. Line passing through zero vector (1 Dimensional)

d. Plane passing though zero vector (2 Dimensional)

e. 3D figure containing zero vector (3 Dimensional)

Read Part 11 : Row Space, Column Space, and Null Space

You can view the complete series here
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