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Calculus and Beyond Homework Help
[Linear Algebra] Linear Transformations, Kernels and Ranges
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[QUOTE="iJake, post: 6003610, member: 645939"] [h2]Homework Statement [/h2] Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range. a) ## T : ℝ → ℝ^2, T(x) = (x,x)## b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)## c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ## d) ## T: C[a,b] → ℝ, T(f) = f(a)## e) ## T: C[a,b] → C[a,b], T(f) = f^2## [h2]Homework Equations[/h2] [/B] Transformations are linear if ##T(a+b) = T(a) + T(b)## and if ##T(c \cdot a) = c \cdot T(a)## ##Ker(T) = \{T(x) = 0\}## ##Im(T) = \{T(x) \in W | x \in V\}##[h2]The Attempt at a Solution[/h2] [/B] a) ##T((x,y,z) + (a,b,c)) = ((y-x) + (b-a), (z+y) + (c+b)) = (y-x, z+y) + (b-a, c+b) = T(x,y,z) + T(a,b,c)## ##T(c \cdot (x,y,z)) = T((cx, cy, cz)) = (cy-cx, cz+cy) = (c \cdot (y-x), c \cdot (z+y)) = c \cdot (y-x, z+y) = c\cdot T(x,y,z)## T is linear. ##Ker(T) = 0## ##Im(T) = \{(a,a) | a \in ℝ\} = <(1,1)>## b) ##T((x,y,z) + (a,b,c)) = ((y-x) + (b-a), (z+y)+(c+b)) = (y-x, z+y) + (b-a, c+b) = T(x,y,z) + T(a,b,c)## ##T(c \cdot (x,y,z)) = T((cx,cy,cz)) = (cy-cx, cz+cy) = (c \cdot (y-x), c \cdot (z+y)) = c \cdot (y-x, z+y) = c \cdot T(x,y,z)## T is linear. I will try to save some space. ##Ker(T) = \{(x,y,z) \in \mathbb R^3 | T(x,y,z) = (0,0,0)\}## ##Ker(T) = \{(y,y,-y) | y \in ℝ\} = <(1,1,-1)>## ##Im(T) = y(1,1) + x(1,0) + z(0,1) = <(1,1), (1,0), (0,1)>## c) ##T((x,y,z) + (a,b,c))## holds, but ##T(c \cdot (x,y,z))## does not hold. I am not sure if the correct notation would be that it works out to ##c^2 \cdot T(x) + c \cdot T(y,z)## but in any case it works out to a non-linear transformation. ##Ker(T) = \{(x,y,z) | (x^2, x, z-x) = (0,0,0)\}## ##(x,y,z) = (0,y,0) | y \in ℝ## ##Ker(T) = \{(0,y,0) | y \in ℝ\} = <(0,1,0)>## ##Im(T)## is not linear. d) ##T(f+g) = (f(a) + g(a)) = f(a) + g(a) = T(f) + T(g)## ##T(c \cdot f) = (c \cdot f)(a) = c \cdot f(a) = c \cdot T(f)## T is linear. ##Ker(T) = \{f \in C[a,b] | T(f) = 0, f \in C[a,b] | f(a) = 0\}## ##Im(T) = \{r \in ℝ | r = f(a), f \in C[a,b]\}## e) ##T(f+g) = (f^2 + g^2) = f^2 + g^2 = T(f) + T(g)## ##T(c \cdot f) = (c \cdot f)^2 = c^2 \cdot f^2 = c^2 \cdot T(f)## T is not linear. ##Ker(T) = \{f \in C[a,b] | f^2 = 0\}## ##Im(T)## is not linear. [/QUOTE]
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[Linear Algebra] Linear Transformations, Kernels and Ranges
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