If
is an inner product on a vector space V, and v is any vector in V, then
defined by
is a linear transformation.

Question

If 
  is an inner product on a vector space V, and v is any vector in V, then 
  defined by 
  is a linear transformation.

Quizplus · Accepted Answer

To show that 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00 is a linear transformation, we need to verify that it satisfies the two properties of linearity:

1. Additivity: 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(u+v) = 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(u) + 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(v) for all vectors u,v in V.
2. Homogeneity: 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(cu) = c11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(u) for all vectors u in V and all scalars c.

Let's check these two properties:

1. For any vectors u,v in V, we have:

11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(u+v) = 11edaba5_20b8_0e6c_a975_e1aa02eb2d93_TBMC1039_00(u+v, v)

= 11edaba5_20b8_0e6c_a975_e1aa02eb2d93_TBMC1039_00(u, v) + 11edaba5_20b8_0e6c_a975_e1aa02eb2d93_TBMC1039_00(v, v)  (by the linearity of the inner product)

= 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(u) + 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(v)

Therefore, 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00 satisfies additivity.

2. For any vector u in V and any scalar c, we have:

11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(cu) = 11edaba5_20b8_0e6c_a975_e1aa02eb2d93_TBMC1039_00(cu, v)

= c * 11edaba5_20b8_0e6c_a975_e1aa02eb2d93_TBMC1039_00(u, v)  (by the linearity of the inner product)

= c * 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00(u)

Therefore, 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00 satisfies homogeneity.

Since 11edaba5_20b8_0e6e_a975_d3875f3b695b_TBMC1039_00 satisfies both properties of linearity, it is indeed a linear transformation.

If Is an Inner Product on a Vector Space