a)Let v be a fixed vector in R^3. Show that the transformation defined by T(u)=vxu is a linear transformation.
b)Find the range of this linear transformation.
Thanx
a) To show that the transformation T(u) = v × u is a linear transformation, we need to prove two properties: additivity and scalar multiplication.
Additivity property:
Let u1 and u2 be two vectors in R^3. We want to show that T(u1 + u2) = T(u1) + T(u2).
Using the definition of T, we have T(u1 + u2) = v × (u1 + u2).
Now, expand the cross product:
v × (u1 + u2) = v × u1 + v × u2.
Since the cross product is distributive, we have:
v × (u1 + u2) = v × u1 + v × u2 = T(u1) + T(u2).
Therefore, the additivity property holds.
Scalar multiplication property:
Let c be a scalar. We want to show that T(cu) = cT(u).
Using the definition of T, we have T(cu) = v × (cu).
Now, expand the cross product:
v × (cu) = c(v × u).
Since the cross product is associative with scalar multiplication, we have:
v × (cu) = c(v × u) = cT(u).
Therefore, the scalar multiplication property holds.
Since both the additivity and scalar multiplication properties hold, we can conclude that T(u) = v × u is a linear transformation.
b) To find the range of this linear transformation, we need to determine the set of all possible outputs (images) of the transformation T.
Since T(u) = v × u, the range of this transformation consists of all vectors that can be expressed as v × u for some u.
Each cross product v × u is perpendicular to both v and u. Therefore, the range of this linear transformation is the set of all vectors in R^3 that are orthogonal to v. In other words, the range of this transformation is the orthogonal complement of the vector v.
To summarize:
Range of T(u) = v × u = set of all vectors orthogonal to v.