(a) State the rank-nullity theorem for a linear map between finite-dimensional vector…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch The four subspaces and the Fredholm alternative

Problem

(a) State the rank-nullity theorem for a linear map T:V→WT:V\to W between finite-dimensional vector spaces, and prove it by extending a basis of N(T)\mathcal{N}(T) to a basis of VV. (b) The matrix A=(120124153616)\mathsf{A} = \begin{pmatrix}1&2&0&1\\2&4&1&5\\3&6&1&6\end{pmatrix} represents a linear map R4→R3\mathbb{R}^4\to\mathbb{R}^3. Find a basis for each of the four fundamental subspaces N(A)\mathcal{N}(\mathsf{A}), R(A)\mathcal{R}(\mathsf{A}), R(AT)\mathcal{R}(\mathsf{A}^{\mathrm{T}}) and N(AT)\mathcal{N}(\mathsf{A}^{\mathrm{T}}). (c) Check the two dimension counts that the rank-nullity theorem gives, one for A\mathsf{A} and one for AT\mathsf{A}^{\mathrm{T}}, and verify by direct dot products that N(A)⊥R(AT)\mathcal{N}(\mathsf{A})\perp\mathcal{R}(\mathsf{A}^{\mathrm{T}}) and N(AT)⊥R(A)\mathcal{N}(\mathsf{A}^{\mathrm{T}})\perp\mathcal{R}(\mathsf{A}).

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