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Created by M. Oki Orlandofrom the Noun Project

Matrix of Linear Map

Definition (Matrix of Linear Map)

Suppose that V,WV,W are finite dimensional vector spaces, where {v1,...,vn}\{v_1,...,v_n\} is a basis for VV and {w1,...,wm}\{w_1,...,w_m\} is a basis for WW. Also suppose that T∈L(V,W)T\in\mathscr{L}(V,W) is a linear map. For j=1,...,nj=1,...,n we can write T(vj)=a1,jw1+...+am,jwmT(v_j)=a_{1,j}w_1+...+a_{m,j}w_m We define the Matrix of Linear Map TT to be M(T):=[ai,j]=[a1,1a1,2⋯a1,na2,1⋱⋮⋮⋱⋮am,1⋯⋯am,n]\mathcal{M}(T):=[a_{i,j}]= \begin{bmatrix} a_{1,1} & a_{1,2} &\cdots & a_{1,n} \\ a_{2,1} & \ddots & &\vdots \\ \vdots & & \ddots & \vdots \\ a_{m,1} & \cdots & \cdots & a_{m,n}\\ \end{bmatrix}

Remark

There is another way to understand what M(T)\mathcal{M}(T) is. We have shown that any two vector spaces with the same dimension are isomorphic, meaning V≅RnV\cong\mathbb{R}^n and W≅RmW\cong\mathbb{R}^m. Set {e1,...,en}\{e_1,...,e_n\} to be the standard basis for Rn\mathbb{R}^n and {f1,...,fm}\{f_1,...,f_m\} to be the standard basis of Rm\mathbb{R}^m. There is an isomorphism \upphiV:Rn→V\upphi_V:\mathbb{R}^n\to V such that \upphiV(ei)=vi\upphi_V(e_i)=v_i for i=1,...,ni=1,...,n and another isomorphism \upphiW:Rm→V\upphi_W:\mathbb{R}^m\to V such that \upphiW(fi)=wi\upphi_W(f_i)=w_i. The composition \upphiW−1∘T∘\upphiV:Rn→Rm\upphi^{-1}_W\circ T\circ\upphi_V:\mathbb{R}^n\to\mathbb{R}^m is a linear map, and can be represented by a matrix. That matrix is M(T)\mathcal{M}(T). Rn→M(T)Rm;SV↓↑SW−1V→TW \begin{CD} \mathbb{R}^n @>{\mathscr{M}(T)}>> \mathbb{R}^m; \\ @V{{S_V}}VV @AA{S_W^{-1}}A \\ V @>>{T}> W\\ \end{CD}

Remark

This essentially proves that M\mathcal{M} is a linear map.

Proposition (Linearity properties for matrices of linear maps)

Let V,WV,W be finite dimensional vector spaces, with bases {v1,...,vn}⊂V\{v_1,...,v_n\}\subset V and {w1,...,vm}⊂W\{w_1,...,v_m\}\subset W. If we have linear maps S,T∈L(V,W)S,T\in\mathscr{L}(V,W), then M(S+T)=M(S)+M(T)\mathcal{M}(S+T)=\mathcal{M}(S)+\mathcal{M}(T) If λ∈F\lambda\in\mathbb{F}, then M(λT)=λM(T)\mathcal{M}(\lambda T)=\lambda \mathcal{M}(T)

Proposition (Linear maps are isomorphic to their matrices)

If V,WV,W are finite dimensional vector spaces with dim(V)=ndim(V)=n and dim(W)=mdim(W)=m. Then the linear maps are isomorphic to their matrices L(V,W)≅Fm×n\mathscr{L}(V,W)\cong\mathbb{F}^{m\times n}

Proposition (Composition rules for matrices)

Let V,W,UV,W,U be vector spaces and let {v1,...,vn},{w1,...,wm}\{v_1,...,v_n\},\{w_1,...,w_m\} and {u1,...,up}\{u_1,...,u_p\} be bases for V,W,UV,W,U respectively. If T∈L(V,W)T\in\mathscr{L}(V,W) and S∈L(W,U)S\in\mathscr{L}(W,U) then M(S∘T)=M(S)M(T)\mathcal{M}(S\circ T) = \mathcal{M}(S)\mathcal{M}(T)

Proposition (Inverse property of matrix of linear map)

If T∈L(V,W)T\in\mathscr{L}(V,W) is an invertible linear map, then the matrix of T−1T^{-1} is M(T−1)=M(T)−1.\mathcal{M}(T^{-1})=\mathcal{M}(T)^{-1}.

Linked from

Matrix of Linear Map

Rank

Upper-Triangular