let K be Field; :: thesis: for V, W being finite-dimensional VectSp of K

for A being Subset of V

for B being Basis of V

for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let V, W be finite-dimensional VectSp of K; :: thesis: for A being Subset of V

for B being Basis of V

for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let A be Subset of V; :: thesis: for B being Basis of V

for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let B be Basis of V; :: thesis: for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let T be linear-transformation of V,W; :: thesis: for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let l be Linear_Combination of B \ A; :: thesis: ( A is Basis of (ker T) & A c= B implies T . (Sum l) = Sum (T @* l) )

assume ( A is Basis of (ker T) & A c= B ) ; :: thesis: T . (Sum l) = Sum (T @* l)

then A1: T | (B \ A) is one-to-one by RANKNULL:22;

A2: (T | (B \ A)) | (Carrier l) = T | (Carrier l) by RELAT_1:74, VECTSP_6:def 4;

then A3: T | (Carrier l) is one-to-one by A1, FUNCT_1:52;

consider G being FinSequence of V such that

A4: G is one-to-one and

A5: rng G = Carrier l and

A6: Sum l = Sum (l (#) G) by VECTSP_6:def 6;

set H = T * G;

A7: rng (T * G) = T .: (Carrier l) by A5, RELAT_1:127

.= Carrier (T @* l) by A3, ZMODUL05:56 ;

dom T = [#] V by RANKNULL:7;

then T * G is one-to-one by A1, A2, A4, A5, FUNCT_1:52, RANKNULL:1;

then A8: Sum (T @* l) = Sum ((T @* l) (#) (T * G)) by A7, VECTSP_6:def 6;

T * (l (#) G) = (T @* l) (#) (T * G) by A3, A5, ZMODUL05:55;

hence T . (Sum l) = Sum (T @* l) by A6, A8, MATRLIN:16; :: thesis: verum

for A being Subset of V

for B being Basis of V

for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let V, W be finite-dimensional VectSp of K; :: thesis: for A being Subset of V

for B being Basis of V

for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let A be Subset of V; :: thesis: for B being Basis of V

for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let B be Basis of V; :: thesis: for T being linear-transformation of V,W

for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let T be linear-transformation of V,W; :: thesis: for l being Linear_Combination of B \ A st A is Basis of (ker T) & A c= B holds

T . (Sum l) = Sum (T @* l)

let l be Linear_Combination of B \ A; :: thesis: ( A is Basis of (ker T) & A c= B implies T . (Sum l) = Sum (T @* l) )

assume ( A is Basis of (ker T) & A c= B ) ; :: thesis: T . (Sum l) = Sum (T @* l)

then A1: T | (B \ A) is one-to-one by RANKNULL:22;

A2: (T | (B \ A)) | (Carrier l) = T | (Carrier l) by RELAT_1:74, VECTSP_6:def 4;

then A3: T | (Carrier l) is one-to-one by A1, FUNCT_1:52;

consider G being FinSequence of V such that

A4: G is one-to-one and

A5: rng G = Carrier l and

A6: Sum l = Sum (l (#) G) by VECTSP_6:def 6;

set H = T * G;

A7: rng (T * G) = T .: (Carrier l) by A5, RELAT_1:127

.= Carrier (T @* l) by A3, ZMODUL05:56 ;

dom T = [#] V by RANKNULL:7;

then T * G is one-to-one by A1, A2, A4, A5, FUNCT_1:52, RANKNULL:1;

then A8: Sum (T @* l) = Sum ((T @* l) (#) (T * G)) by A7, VECTSP_6:def 6;

T * (l (#) G) = (T @* l) (#) (T * G) by A3, A5, ZMODUL05:55;

hence T . (Sum l) = Sum (T @* l) by A6, A8, MATRLIN:16; :: thesis: verum