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Subspace

Description

A subspace is any subset of \(R_n\) which is also a vector space over \(R_n\).

When \(V\) (a random subset) is a subspace of \(R_n\), it means:

  • \(V\) contains the zero vector
  • If \(\vec{x}\) is in \(V\), for every scalar of it, the result is in \(V\) also
  • if \(\vec{a}\) is in \(V\) and \(\vec{b}\) is in \(V\), the addition of them is in \(V\) also

Basis

A basis for a subspace \(S\) of \(R_n\) is a set of vectors in \(S\) that is linearly independent and is maximal with this property (that is, adding any other vector in \(S\) to this subset makes the resulting set linearly dependent).

  • A space can have multiple basis
  • A basis should be linearly independent

  • Two units \(i\) and \(j\) are standard basis of \(R_2\)

    \(\mathbf{e_x} = (1, 0), \quad \mathbf{e_y} = (0, 1).\)

    Similarly, the standard basis for the three-dimensional space \(\mathbb{R}^3\) is formed by vectors

    \(\mathbf{e_x} = (1, 0, 0), \quad \mathbf{e_y} = (0, 1, 0), \quad \mathbf{e_z} = (0, 0, 1).\)

Kernel

The kernel of a linear map, also known as the null space, is the linear subspace of the domain of the map which is mapped to the zero vector.

Kernel and image of a linear map \(L\) from \(V\) to \(W\):