H_
Homomorphisms
SKIP_ p1 QM_[0;linear map;linear map;not a linear map;linear map;not a linear map;not a linear map] Which of the following spaces are necessarily linear maps? AH_[2]

1. AS_[linear map;not a linear map] h((x, y, z)) = (2x+y, 2z+x-y)
2. AS_[linear map;not a linear map] The map D from the rational functions of one variable into the rational functions of one variable where D(f) = f', i.e. D(f) is the derivative of f.
3. AS_[linear map;not a linear map] G((x,y)) = (0, 1)
4. AS_[linear map;not a linear map] The map from the polynomials to the polynomials defined by .
5. AS_[linear map;not a linear map] The map from the polynomials to the polynomial defined by g(p(x)) = p(x+1)p(x-1)
6. AS_[linear map;not a linear map] The determinant map det(a,b,c,d) = ad - bc.

SKIP_ p2 QM_[0;true;false;true;false] Let P be the vector space of polynomials of one variable with real coefficients. AH_[2] Which are the following are true? Be sure that you can either prove the assertion or give a counterexample.

1. AS_[true;false] The map D which maps every polynomial to its derivative is a linear map.
2. AS_[true;false] The map D has an inverse.
3. AS_[true;false] There is a linear map R from P to P such that D(R(p(x)) = p(x) for every polynomial p(x) in P. (If it exists, what is it?)
4. AS_[true;false] There is a linear map L from P to P such that L(D(p(x)) = p(x) for every polynomial p(x) in P. (If it exists, what is it?)

SKIP_ p3 QM_[0;true;true;true;true;false] Consider the vector space . Be sure that you can either prove the assertion or give a counterexample: AH_[2]

1. AS_[true;false] The perpendicular projection of the vector (a, b, c) into the xy-plane is a linear map. (The map is p((a, b, c)) = (a, b).)
2. AS_[true;false] Let be any vector of unit length. The map f which maps to the vector (where the dot indicates dot product) is a linear map. This is called the component of in the direction .
3. AS_[true;false] Letting be as in the last part, the map p defined by is a linear map. It is called the perpendicular projection into the plane perpendicular to .
4. AS_[true;false] The map p of the first part is the perpendicular projection into the plane perpendicular to (0, 0, 1).
5. AS_[true;false] The map p of the first part cannot be the perpendicular projection into the plane perpendicular to any unit length vector other than (0, 0, 1).

SKIP_ p4 QM_[0;false;true;false;false;true;false;true] Let f be a linear map from a vector space V to a vector space W. Let S be a set of vectors in a vector space V, and f(S) be the set . AH_[2] Which of the following are true? Make sure you can either prove the assertion or give a counter-example.

1. AS_[true;false] If S is linearly independent, then so is f(S).
2. AS_[true;false] If f(S) is linearly independent, then so is S.
3. AS_[true;false] If S spans V, then f(S) spans W.
4. AS_[true;false] If f(S) spans W, then S spans V
5. AS_[true;false] If S spans V, then f(S) spans the rangespace of f.
6. AS_[true;false] If S is a basis of V, then f(S) is a basis of W.
7. AS_[true;false] If f is non-singular (i.e. one-to-one), and f(S) spans W, then S spans V.

SKIP_ p5 QM_[0;true;false;true;false;6;0] Consider the function f from the polynomials of degree at most 5 to the polynomials of at most 7 defined by Answer the following are necessarily true. Make sure you can either prove the assertion or give a counter-example. AH_[2]

1. AS_[true;false] The function f is a linear map.
2. AS_[true;false] The rangespace of f is the set of all polynomials of degree at most 7 with roots at 1.
3. AS_[true;false] The map f is non-singular.
4. AS_[true;false] The map f is onto.
5. The rank of f is AS_[0;1;2;3;4;5;6;7]
6. The nullity of f is AS_[0;1;2;3;4;5;6;7]

SKIP_ p6 QM_[0;true;false;4;true;2] Let f be a linear map from a vector space of dimension 4 to a vector space of dimension 7. Answer the following questions; as usual, you should be able to prove the assertion or give a counter-example. AH_[2]

1. AS_[true;false] The function f cannot be onto.
2. AS_[true;false] The function f cannot be one-to-one.
3. If the nullity is zero, then the rank must be AS_[0;1;2;3;4;5;6;7]
4. AS_[true;false] If the rank is 4, then the nullity is zero.
5. If the rank is 2, then the nullity is AS_[0;1;2;3;4;5;6;7] . SKIP_ p7 QM_[0;true;true;true;true;true] Let f be a linear map from a vector space V to a vector space W. Let S be a basis of the nullspace K of f. Let T be disjoint from S and such that the union of S and T is a basis of V. AH_[2]
   1. AS_[true;false] V is the sum of K and the span of T.
   2. AS_[true;false] V is the direct sum of K and the span of T.
   3. AS_[true;false] f(S) is the trivial subspace of the codomain of f.
   4. AS_[true;false] f(T) is a basis of the rangespace of f.
   5. AS_[true;false] The restriction of f to the span of T is an isomorphism between the span of T and the rangespace of f.
   SKIP_ p8 QM_[0.001;1;true;true;true;false] Let V be a vector space and S be a basis of V. AH_[2] Make sure you can either prove the assertion or give a counterexample:
   1. For every s in S, let be the linear map from V to defined by and for all t in S other than s. The rank of is AC_[7] .
   2. AS_[true;false] is linearly independent.
   3. AS_[true;false] There is a unique linear map g from V to defined by and this linear map is one-to-one.
   4. AS_[true;false] If V is of finite dimension, then the linear map g is an isomorphism.
   5. AS_[true;false] The function g is always an isomorphism.
   SKIP_ p9 QM_[0;true] A linear map from a finite dimensional vector space to itself is one-to-one if and only if it is onto. AS_[true;false] SKIP_ p10 QM_[0;10] What is the dimension of the space of linear maps from a vector space of dimension 5 to a vector space of dimension 2? Try to generalize from problem 8. Can you generalize further? AC_[7] . SKIP_