Question

3. (a) Prove that if W1 and W2 are finite-dimensional subspaces of a vector space V, then the subspace W1 + W2 is finite dimensional and dim(W1 + W2) = dim(W1) + dim(W2) – dim(W1 ? W2). (Hint: Start with a basis {u1, . . . , uk} for W1 ? W2 and extend this set to a basis {u1, . . . , uk, v1, . . . , vm} for W1 and to a basis {u1, . . . , uk, w1, . . . , wn} for W2.) (b) Find an example of subspaces W1 ? {0} and W2 ? {0} of R3 such that: dim(W1 + W2) = dim(W1) + dim(W2). (c) Find an example of subspaces W1 and W2 of R3 such that both dim(W1 + W2) < dim(W1) + dim(W2) and dim(W1 ? W2) < min{dim(W1), dim(W2)}. 4. Fix a field F and an integer n ? 1. For each A ? Mnxn(F), the trace of A, denoted tr(A), is the sum of all diagonal entries akk of A, 1 ? k ? n. Define T : Mnxn(F) ? F by T(A) = tr(A). (a) Prove that T is a linear transformation. (b) Find a basis for the null space N(T). (c) Determine rank(T) and nullity(T) and verify that their sum is dim(Mnxn(F)) (according to the Rank-Nullity Theorem).

          3. (a) Prove that if W1 and W2 are finite-dimensional subspaces of a vector space V, then the subspace W1 + W2 is finite dimensional and
dim(W1 + W2) = dim(W1) + dim(W2) – dim(W1 ? W2).
(Hint: Start with a basis {u1, . . . , uk} for W1 ? W2 and extend this set to a basis {u1, . . . , uk, v1, . . . , vm} for W1 and to a basis {u1, . . . , uk, w1, . . . , wn} for W2.)
(b) Find an example of subspaces W1 ? {0} and W2 ? {0} of R3 such that:
dim(W1 + W2) = dim(W1) + dim(W2).
(c) Find an example of subspaces W1 and W2 of R3 such that both dim(W1 + W2) < dim(W1) + dim(W2) and dim(W1 ? W2) < min{dim(W1), dim(W2)}.
4. Fix a field F and an integer n ? 1. For each A ? Mnxn(F), the trace of A, denoted tr(A), is the sum of all diagonal entries akk of A, 1 ? k ? n. Define T : Mnxn(F) ? F by T(A) = tr(A).
(a) Prove that T is a linear transformation.
(b) Find a basis for the null space N(T).
(c) Determine rank(T) and nullity(T) and verify that their sum is dim(Mnxn(F)) (according to the Rank-Nullity Theorem).

3. (a) Prove that if W1 and W2 are finite-dimensional subspaces of a vector space V, then the subspace W1 + W2 is finite dimensional and
dim(W1 + W2) = dim(W1) + dim(W2) – dim(W1 ? W2).
(Hint: Start with a basis u1, . . . , uk for W1 ? W2 and extend this set to a basis u1, . . . , uk, v1, . . . , vm for W1 and to a basis u1, . . . , uk, w1, . . . , wn for W2.)
(b) Find an example of subspaces W1 ? 0 and W2 ? 0 of R3 such that:
dim(W1 + W2) = dim(W1) + dim(W2).
(c) Find an example of subspaces W1 and W2 of R3 such that both dim(W1 + W2) < dim(W1) + dim(W2) and dim(W1 ? W2) < mindim(W1), dim(W2).
4. Fix a field F and an integer n ? 1. For each A ? Mnxn(F), the trace of A, denoted tr(A), is the sum of all diagonal entries akk of A, 1 ? k ? n. Define T : Mnxn(F) ? F by T(A) = tr(A).
(a) Prove that T is a linear transformation.
(b) Find a basis for the null space N(T).
(c) Determine rank(T) and nullity(T) and verify that their sum is dim(Mnxn(F)) (according to the Rank-Nullity Theorem).

Added by Tammy W.

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