Vector Space

The idea of vector space is that it is closed under vector addition and scaler multiplication i.e linear combination

Formally ,

Quick Note

Technically, V is a real vector space. All of the theory also holds for a complex vector space in which the scalars are complex numbers.

Subspace

The main idea is that subspace is a space inside $R^n$ which is also a vector space i.e satisfies the condition of closedness under linear combination

Formally,

Question

• Why is the condition of having 0 vector necessary ?

Null Space / Kernel

In determinant , we saw that when determinant is 0 a lot of vector gets mapped to 0 vector via the linear transformation. The set of all these vectors compound to null space .