# Thread: T maps Rn into Rm and x =x0 +tv be a line in Rn

1. ## T maps Rn into Rm and x =x0 +tv be a line in Rn

Okay I am struggling again, I I think I'm in over my head, a pre-req course would have been helpful. Anyway here I go.

Tell whether the statement is true or false and justify the answer.

a.) if T maps Rn into Rm and T(0) = 0, then T is linear. I say true because it is a zero tranformation. Am I right?

b.) If T maps Rn into Rm, and if T(c1u +c2v) =c1T(u) + c2T(v) for all scalers c1 and c2 and for all vectors u and v, then T is linear. True as this represents the addivtivity and is alinear transformation by definition. Am I right?

c. ) There is only one linear transformtion T : Rn ->Rn such that T(-v) = -T(v) in Rn. Not sure if this is true or false.

d.) there is only one linear transformation T: Rn -> Rn for which T (u + v) = T (u-v) for all vectors u and V in Rn. ???

e.) if vo is a nonzero vector in Rn, then the formula T(v) = v0 +v defines a linear operator in V. Still nto understanding the meaning of linear operator. Help!

2. Originally Posted by cculley3
a.) if T maps Rn into Rm and T(0) = 0, then T is linear. I say true because it is a zero tranformation. Am I right?
No. Consider $T:\mathbb{R}\mapsto \mathbb{R}$ defined as $T(x) = x^2$.

b.) If T maps Rn into Rm, and if T(c1u +c2v) =c1T(u) + c2T(v) for all scalers c1 and c2 and for all vectors u and v, then T is linear. True as this represents the addivtivity and is alinear transformation by definition. Am I right?
Yes, that is the definition of linearity.

c. ) There is only one linear transformtion T : Rn ->Rn such that T(-v) = -T(v) in Rn. Not sure if this is true or false.
Any linear transformation $T:\mathbb{R}^n \mapsto \mathbb{R}^m$ has this property.

d.) there is only one linear transformation T: Rn -> Rn for which T (u + v) = T (u-v) for all vectors u and V in Rn. ???
Yes. If $T(\bold{u}+\bold{v}) = T(\bold{u} - \bold{v})$ it means $T(\bold{u}) + T(\bold{v}) = T(\bold{u}) - T(\bold{v})$ thus $T(\bold{v}) = - T(\bold{v})$ thus $T(\bold{v}) = 0$ thus $T$ is the zero transformation.

e.) if vo is a nonzero vector in Rn, then the formula T(v) = v0 +v defines a linear operator in V. Still nto understanding the meaning of linear operator. Help!
Is it true that $T(a\bold{u}+b\bold{v}) = aT(\bold{u}) + bT(\bold{v})$?

3. ## T maps Rn into Rm

Originally Posted by ThePerfectHacker
No. Consider $T:\mathbb{R}\mapsto \mathbb{R}$ defined as $T(x) = x^2$.

Yes, that is the definition of linearity.

Any linear transformation $T:\mathbb{R}^n \mapsto \mathbb{R}^m$ has this property.

Yes. If $T(\bold{u}+\bold{v}) = T(\bold{u} - \bold{v})$ it means $T(\bold{u}) + T(\bold{v}) = T(\bold{u}) - T(\bold{v})$ thus $T(\bold{v}) = - T(\bold{v})$ thus $T(\bold{v}) = 0$ thus $T$ is the zero transformation.

Is it true that $T(a\bold{u}+b\bold{v}) = aT(\bold{u}) + bT(\bold{v})$?
Yes this true, there fore this statement is true, correct?