Download Álgebra Lineal y Algunas de sus Aplicaciones by L.I. GOLOVINA PDF

Download Álgebra Lineal y Algunas de sus Aplicaciones by L.I. GOLOVINA PDF By L.I. GOLOVINA

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Extra info for Álgebra Lineal y Algunas de sus Aplicaciones

Example text

Then, if we represent vectors of V and W as column vectors relative to these bases, we have α(v) = Av. Proof Let e1 , . . , em be the basis for V , and f1 , . . , fn for W . Take v = ∑m i=1 ci ei ∈ V , so that in coordinates   c1 .  v = ..  . cm Then m m α(v) = ∑ ci α(ei ) = ∑ i=1 n ∑ cia ji f j , i=1 j=1 so the jth coordinate of α(v) is ∑ni=1 a ji ci , which is precisely the jth coordinate in the matrix product Av. In our example, if v = 2e1 + 3e2 + 4e3 = [ 2 3 4 ] , then   2 1 2 3   20 α(v) = Av = 3 = .

1 Let V and W be vector spaces over a field K. A function α from V to W is a linear map if it preserves addition and scalar multiplication, that is, if • α(v1 + v2 ) = α(v1 ) + α(v2 ) for all v1 , v2 ∈ V ; • α(cv) = cα(v) for all v ∈ V and c ∈ K. Remarks 1. We can combine the two conditions into one as follows: α(c1 v1 + c2 v2 ) = c1 α(v1 ) + c2 α(v2 ). 2. In other literature the term “linear transformation” is often used instead of “linear map”. 2 Let α : V → W be a linear map. The image of α is the set Im(α) = {w ∈ W : w = α(v) for some v ∈ V }, and the kernel of α is Ker(α) = {v ∈ V : α(v) = 0}.

Take v = ∑m i=1 ci ei ∈ V , so that in coordinates   c1 .  v = ..  . cm Then m m α(v) = ∑ ci α(ei ) = ∑ i=1 n ∑ cia ji f j , i=1 j=1 so the jth coordinate of α(v) is ∑ni=1 a ji ci , which is precisely the jth coordinate in the matrix product Av. In our example, if v = 2e1 + 3e2 + 4e3 = [ 2 3 4 ] , then   2 1 2 3   20 α(v) = Av = 3 = . 1 5 −1 13 4 Addition and multiplication of linear maps correspond to addition and multiplication of the matrices representing them. 5 Let α and β be linear maps from V to W .