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A matrix is a linear transformation if, T(u+v)= T(u) +T(v) and T(cu)=cT(u).

Theorem 8.4.2 If V is a finnite dimensional vector space, and T: V-> V is a linear operator then the following are equivalent.

a) T is one to one, b) ker(T)=0, c) nuttility(T)=0 and d) the Range of T is V; R(T)=V

1- Question

Let V be a fixed vector in R^3. a)Show that the transformation defined by T(u)=vXuis a linear transformation.

b) Find the range ot the linear transformation

c) If v=i , find the matrix for this linear transformation.

2- Answer

a) I proved that T(u+v)= T(u)+T(v) and T(cu)=cT(u), through cross product properties, And therefore proved its linear transformation.

b) This one i am not entirely sure, however with the Theroem 8.4.2 d) I concluded that R(T) = R^3

c) This one is where I am unsure where to begin. Since T(u) =vXu

Then T(u) = { Det ( v2, v3 - Det ( v1, v3 Det ( v1, v2

u2, u3 ) , u1, u3) , u1, u2) }

I am not sure where to go from there, and unsure what they mean by v=i

Thank you,

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# Linear Algebra; Transformation of cross product

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