- This Is For Linear Algebra 1 (38.43 KiB) Viewed 46 times
This is for Linear Algebra
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answerhappygod
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This is for Linear Algebra
This is for Linear Algebra
Problem 6. Suppose T: V→ V is a linear operator, p(x) = (x-2)(x-3), and suppose p(T) = 0. Let V₂ = V3 = ker(T-21) ker(T-31) Show that I acts on V₂ as multiplication by 2 and on V3 as multiplication by 3. Find scalars a and 3 so a(x-2) + 3(x-3) = 1 and show that the maps P= 3(T-31) Q = a(T-21) have P+Q=I. (In the above, I : V→V is the identity operator). Show that the image of P is in V₂ while the image of Q is in V3. Show that any vector v EV can be written as v = V2 + V3 where v2 = P(v) and v3 = Q(v).
Problem 6. Suppose T: V→ V is a linear operator, p(x) = (x-2)(x-3), and suppose p(T) = 0. Let V₂ = V3 = ker(T-21) ker(T-31) Show that I acts on V₂ as multiplication by 2 and on V3 as multiplication by 3. Find scalars a and 3 so a(x-2) + 3(x-3) = 1 and show that the maps P= 3(T-31) Q = a(T-21) have P+Q=I. (In the above, I : V→V is the identity operator). Show that the image of P is in V₂ while the image of Q is in V3. Show that any vector v EV can be written as v = V2 + V3 where v2 = P(v) and v3 = Q(v).