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Math Help - T maps Rn into Rm and x =x0 +tv be a line in Rn

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by cculley3 View Post
    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})?
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  3. #3
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    T maps Rn into Rm

    Quote Originally Posted by ThePerfectHacker View Post
    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?
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