Page 75 - Linear Algebra Done Right

Exercises
m
n
Suppose T ∈L(F , F ) and that
19.
that T has the form

promised on page 39.
a 1,1 ... a 1,n  This exercise shows 61
 . . 
M(T) =  . . . .  ,
 
a m,1 ... a m,n
where we are using the standard bases. Prove that
T(x 1 ,...,x n ) = (a 1,1 x 1 +· · ·+a 1,n x n ,...,a m,1 x 1 +· · ·+a m,n x n )
n
for every (x 1 ,...,x n ) ∈ F .
20. Suppose (v 1 ,...,v n ) is a basis of V. Prove that the function
T : V → Mat(n, 1, F) defined by

Tv =M(v)

is an invertible linear map of V onto Mat(n, 1, F); here M(v) is
the matrix of v ∈ V with respect to the basis (v 1 ,...,v n ).

21. Prove that every linear map from Mat(n, 1, F) to Mat(m, 1, F) is
given by a matrix multiplication. In other words, prove that if
T ∈L(Mat(n, 1, F), Mat(m, 1, F)), then there exists an m-by-n
matrix A such that TB = AB for every B ∈ Mat(n, 1, F).

22. Suppose that V is finite dimensional and S, T ∈L(V). Prove that
ST is invertible if and only if both S and T are invertible.

23. Suppose that V is finite dimensional and S, T ∈L(V). Prove that
ST = I if and only if TS = I.

24. Suppose that V is finite dimensional and T ∈L(V). Prove that
T is a scalar multiple of the identity if and only if ST = TS for
every S ∈L(V).

25. Prove that if V is finite dimensional with dim V> 1, then the set
11:45 am, Jan 11, 2005
of noninvertible operators on V is not a subspace of L(V).