Question

2 (15 points) If {v1, v2, ..., vm} spans a vector space V, does {v1 + v2, v2 + v3, ..., vj-1 + vj, ..., vm + v1} spans V as well ? Does the answer depends whether m is odd or even? Prove your answer. 3 (10 points) Define T ? L(?³, ?³) by T(x, y, z) = (x + y + z, x + y + z, x + y + z). Find rank(T) and nullity(T). 4 (10 points) Let U1 and U2 be two subspaces of a vector space V. Suppose that U1 ? U2 is a subspace of V, prove that either U1 ? U2 or U2 ? U1. 5 (10 points) Let ?, ? be the standard ordered bases for ?² and ?³, respectively, a linear transformation T : ?² ? ?³ is defined by T(a1, a2) = (2a1 - a2, 3a1 - 5a2, a2). Find [T]??. 6 (10 points) Suppose that p0, p1, ..., pm are polynomial functions in Pm(?) (the vector space of polynomial functions with degree less than or equal to m) such that pj(2) = 0 for all j = 0, 1, ..., m. Prove that {p0, p1, ..., pm} is not linearly independent in Pm(?). 7 (15 points) Let V be a finite-dimensional vector space, and T : V ? V be a linear transformation. Prove (a) R(T²) ? R(T). (b) If rank(T) = rank(T²), then R(T) ? N(T) = {0}. 8 (10 points) Suppose that V and W are finite-dimensional vector spaces, T ? L(V, W), S ? L(W, Z). Prove that dim N(ST) ? dim N(S) + dim N(T).

          2 (15 points) If {v1, v2, ..., vm} spans a vector space V, does
{v1 + v2, v2 + v3, ..., vj-1 + vj, ..., vm + v1}
spans V as well ? Does the answer depends whether m is odd or even? Prove your answer.

3 (10 points) Define T ? L(?³, ?³) by T(x, y, z) = (x + y + z, x + y + z, x + y + z).
Find rank(T) and nullity(T).

4 (10 points) Let U1 and U2 be two subspaces of a vector space V. Suppose that U1 ? U2 is a subspace of V, prove that either U1 ? U2 or U2 ? U1.

5 (10 points) Let ?, ? be the standard ordered bases for ?² and ?³, respectively, a linear transformation T : ?² ? ?³ is defined by T(a1, a2) = (2a1 - a2, 3a1 - 5a2, a2).
Find [T]??.

6 (10 points) Suppose that p0, p1, ..., pm are polynomial functions in Pm(?) (the vector space of polynomial functions with degree less than or equal to m) such that pj(2) = 0 for all j = 0, 1, ..., m. Prove that {p0, p1, ..., pm} is not linearly independent in Pm(?).

7 (15 points) Let V be a finite-dimensional vector space, and T : V ? V be a linear transformation. Prove
(a) R(T²) ? R(T).
(b) If rank(T) = rank(T²), then R(T) ? N(T) = {0}.

8 (10 points) Suppose that V and W are finite-dimensional vector spaces, T ? L(V, W), S ? L(W, Z). Prove that
dim N(ST) ? dim N(S) + dim N(T).

2 (15 points) If v1, v2, ..., vm spans a vector space V, does
v1 + v2, v2 + v3, ..., vj-1 + vj, ..., vm + v1
spans V as well ? Does the answer depends whether m is odd or even? Prove your answer.

3 (10 points) Define T ? L(?³, ?³) by T(x, y, z) = (x + y + z, x + y + z, x + y + z).
Find rank(T) and nullity(T).

4 (10 points) Let U1 and U2 be two subspaces of a vector space V. Suppose that U1 ? U2 is a subspace of V, prove that either U1 ? U2 or U2 ? U1.

5 (10 points) Let ?, ? be the standard ordered bases for ?² and ?³, respectively, a linear transformation T : ?² ? ?³ is defined by T(a1, a2) = (2a1 - a2, 3a1 - 5a2, a2).
Find [T]??.

6 (10 points) Suppose that p0, p1, ..., pm are polynomial functions in Pm(?) (the vector space of polynomial functions with degree less than or equal to m) such that pj(2) = 0 for all j = 0, 1, ..., m. Prove that p0, p1, ..., pm is not linearly independent in Pm(?).

7 (15 points) Let V be a finite-dimensional vector space, and T : V ? V be a linear transformation. Prove
(a) R(T²) ? R(T).
(b) If rank(T) = rank(T²), then R(T) ? N(T) = 0.

8 (10 points) Suppose that V and W are finite-dimensional vector spaces, T ? L(V, W), S ? L(W, Z). Prove that
dim N(ST) ? dim N(S) + dim N(T).

Added by Nicole A.

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