ANSWERS: 1
  • Find the basis like this: Pick a nice random vector 1,1,1,1 Subtract any component it might have in the given directions: component in the [1,2,-1,0] direction is ([1,2,-1,0]dot[1,1,1,1]) / ([1,2,-1,0]dot[1,2,-1,0]) = 2 / 6 so subtract 2/6 of [1,2,-1,0] from the first vector to give: [2/3,1/3,4/3,1] this is a vector orthoganal to the first vector. repeat the process on the second vector to give a vector orthogonal to both. Now think of a new random vector and repeat the process on the first three vectors to produce a fourth. the last two vectors are a basis for the kernal. And two vectors means dimension = 2. You can get a linear map by simply repeating the above process on an arbitrary vector x.

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