linear algebra - find rank and nullify of matrix transformation?

Let T be a linear transformation from M5,5(R) to P16(R).

What would be the maximum rank, minimum nutility, and maximum nutility? I was wondering how would I find these values just by looking at the dimensions/degree of these matrices/polynomials?

2 Answers

  • 1 month ago
    Favourite answer

    The first step in this problem is to determine the dimensions of both vector spaces. (Note that these are both dimensions with respect to the reals)

    dim(M5,5) = 5*5 = 25

    dim(P16) = 16 + 1 = 17

    The rank of a transformation is the dimension of its range, aka column space or image, of the transformation.

    Because the range is a subset of the codomain (P16 in this case) the rank can be at most the dimension of the codomain, so in this case the maximum rank is 17.

    The nullity of a transformation is the dimension of its null space, or kernel, which is the subset of the domain which maps to the zero vector in the codomain.

    Notice again that the dimension of a subspace can only be at most the dimension of the original space, so the nullity is at most the dimension of the domain, M5,5. So the maximum nullity is 25.

    Finally, for the minimum nullity, recall the rank-nullity theorem: the rank plus the nullity must equal the dimension of the domain. When nullity is at its lowest, rank must be at its highest, so we can plug in the maximum rank of 17 to find the minimum nullity.

    17 + n = 25 => n = 8, so the minimum nullity is 8.

    I hope this helps!

  • 1 month ago

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