3DJH๎๎๎๎
๎๎๎๎. Which of the following are linear transformations of
L:R3รฉR3? If it is, prove it. If it isnโt, exhibit an
example of a property which fails. E.g., L(3
๎
[1, 1, 1]) =
L([3, 3, 3]) = [9, 6, 3], But 3
๎
L([1, 1, 1]) = 2
๎
[3, 2, 1] = [6, 4, 2].
๎D๎ L([x, y, z]) = [ ]
x
,y2+z2,z2
๎E๎ L([x, y, z]) = [1, z, y]
๎F๎ L([x, y, z]) = [0, z, y]
๎๎๎๎. Find the standard matrix A for each L.
๎D๎ L: R3รฉ R2 by L([x, y, z]T) = [x, y]T.
๎E๎ L: R3รฉ R3 by L([x, y, z]T) = r [x, y, z]T. This
transformation stretches or dilates the space by a factor of r.
๎F๎ L: R2รฉ R2 by L([x, y]T) = [x, -y]T. This transformation
reflects the place vertically around the x-axis.
๎๎๎๎๎. Suppose L:R2รฉR2 is a linear transformation such
that L([1, 1]) = [1, -2] and L([-1, 1]) = [2, 3].
๎D๎ L([-1, 5]) =
๎E๎ L([a, b]) =
๎๎๎๎๎. Let w be a fixed vector in an inner product space V.
Let L:VรฉV by L(v) = (v, w). Prove that L is a linear
transformation.
๎๎๎๎๎. Let W be the vector space of all real-valued
functions and let V be the subspace of all differential
functions.
Define L:VรฉW by L( f ) = fรฒ where fรฒ is the derivative of f.
Prove that L is a linear transformation.
Math 311 Hw 23 Name _____________________________ Score_____/ 13
Hw 263: 2abc, 8abc, 12ab, 16, 22. Recommended 263: 1, 3, 9, 11, 15. Answer page 4.1, 545.