Re: [線代] linear transformation
※ 引述《mqazz1 (無法顯示)》之銘言：
You can verify it by the definition of "linear" transformation.
: determine whether the following are linear transformations fron
: C[0, 1] into R^1
Given f,g in C[0,1], a in |R^1
: (a) L(f) = f(0)
L(af+g) = (af+g)(0) = (af)(0) + g(0) = af(0) + g(0) = aL(f) + L(g)
So, it is a linear transformation(functional?)
: (b) L(f) = |f(0)| //絕對值
L(af) = |af(0)| = |a| |f(0)| may not equal to aL(f) if a is negative.
Hence, it is not a linear transformation.
: (c) L(f) = [ f(0)+f(1) ] / 2
Similarly as (a),
L(af+g) = [ (af+g)(0)+(af+g)(1) ] / 2
= a[(f(0)+f(1))/2] + [(g(0)+g(1))/2]
= aL(f)+L(g)
: 1 2 1/2
: (d) L(f) = { ∫ [f(x)] dx }
: 0
1 1/2 1 1/2
L(af) = {∫ [af(x)]^2 dx} = |a| {∫ [af(x)]^2 dx}
0 0
Similarly as (b), it is not a linear transformation.
: 請問這題要怎麼判斷呢?
: 謝謝!!
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05/14 22:26, , 1F
05/14 22:26, 1F
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