Math 560, Assignment 3
Due Monday, October 3
1. Prove Theorem 4.2.3.
2. Prove Theorem 4.3.1.
3. Prove Theorem 4.3.2.
4. Let Tand Sbe linear transformations on a finite dimensional vector space.
(a) Prove that Tand Sare invertible if and only if both ST and T S are invertible.
(b) Prove that if ST =I, then both Sand Tare invertible.
5. (a) Let Xbe the space of polynomials of degree less than nand let Tā L(X) denote second differentiation,
i.e., T=d2/dx2. Compute the matrix associated with T.
(b) Let X= span {ex, e2x,Ā· Ā· Ā· , enx }, which also forms a basis for X. Compute the matrix of T=d/dx ā L(X).
(c) Let Xbe the space of all real two-by-two matrices and let Tbe the linear transformation that sends every
matrix Aonto P A, where P=Āµ1 1
1 1Ā¶. Find the matrix of Awith respect to the basis consisting of
Ā½Āµ1 0
0 0Ā¶,Āµ0 1
0 0Ā¶,Āµ0 0
1 0Ā¶,Āµ0 0
0 1Ā¶Ā¾.
6. What happens to the matrix of a linear transformation on a finite dimensional vector space when elements of
the basis with respect to which the matrix is computed are permuted among themselves?
7. Prove that if {x1,Ā· Ā· Ā· , xk}and {y1,Ā· Ā· Ā· , yk}are linearly independent sets of vectors in a finite dimensional
vector space Xthere is a transformation Tā L(X) such that Txj=yjfor 1 ā¤jā¤k.
8. Let A=ĀµA11 A12
A21 A22Ā¶and B=ĀµB11 B12
B21 B22Ā¶be two matrices in block form with the same dimensions for
themselves and the blocks. Show that
ĀµA11 A12
A21 A22Ā¶ ĀµB11 B12
B21 B22Ā¶=ĀµA11 B11 +A12B21 A11B12 +A12 B22
A21B11 +A22 B21 A21B12 +A22B22 Ā¶.
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