3) The point of this problem is to show that linear maps out of/into direct sums "behave" like matrices. Let V₁,..., Vn,

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3 The Point Of This Problem Is To Show That Linear Maps Out Of Into Direct Sums Behave Like Matrices Let V Vn 1 (243.86 KiB) Viewed 10 times

3) The point of this problem is to show that linear maps out of/into direct sums "behave" like matrices. Let V₁,..., Vn, W₁,..., Wm be vector spaces over F. (a) (2 points) Suppose we are given linear maps Tij : V; → W; for each i,j. Construct a linear map Wm T: V₁ V₂ → W₁ that "behaves" like the mxn matrix of linear maps T11 T12 T21 T22 Tm1 Tm2 V₂ → (b) (2 points) Show that every linear map T: V₁ Wm is of this form for linear maps Tij: V; → W₂ (hint: inclusion into direct sum/projection onto subspace) (c) (1 point) Conclude that the set of linear maps can be identified with the set of matrices T11 T12 T21 T22 Tin T2n → W₁ T: V₁ 0 · · · 0 V₂ → W₁ Wm Tm1 Tm2 Tmn. Tin T2n Tmn with coefficients Tij € L(V₁, W₂) (d) (1 point) Does anything familiar happen when V₁ = ... = V₁ = W₁ = ... = Wm = F? (Hint: when V₁ = W₁ = F what is L(V;, W;) isomorphic to? Think dimensions....)