Knowlpedia

Images and preimages of subspaces under linear maps

Linear maps send subspaces to subspaces and pull back subspaces to subspaces

Proposition. Let T:X→YT:X\to Y be a linear operator between vector spaces.

  1. If M⊂XM\subset X is a , then T(M)⊂YT(M)\subset Y is a linear subspace.
  2. If N⊂YN\subset Y is a linear subspace, then the preimage
    T−1(N):={x∈X∣T(x)∈N}T^{-1}(N):=\{x\in X\mid T(x)\in N\}

is a linear subspace of XX.

Remarks

In particular, ker⁡T\ker T and im⁡T\operatorname{im}T from are subspaces.

Proof sketch. For (1), use T(m1+m2)=T(m1)+T(m2)T(m_1+m_2)=T(m_1)+T(m_2) and T(λm)=λT(m)T(\lambda m)=\lambda T(m) and closure of MM. For (2), if x1,x2∈T−1(N)x_1,x_2\in T^{-1}(N) then T(x1+x2)∈NT(x_1+x_2)\in N by closure of NN; similarly for scalars.