Page 133 - Linear Algebra Done Right

Chapter 6. Inner-Product Spaces
120
products
(ST) = T S for all T ∈L(V, W) and S ∈L(W, U) (here U is an
∗
∗ ∗
inner-product space over F).
The next result shows the relationship between the null space and
the range of a linear map and its adjoint. The symbol ⇐⇒ means “if and
only if”; this symbol could also be read to mean “is equivalent to”.
6.46 Proposition: Suppose T ∈L(V, W). Then
(a) null T ∗ = (range T) ;
⊥
(b) range T ∗ = (null T) ;
⊥
(c) null T = (range T ) ;
∗ ⊥
(d) range T = (null T ) .
∗ ⊥
Proof: Let’s begin by proving (a). Let w ∈ W. Then

w ∈ null T ∗ ⇐⇒ T w = 0
∗
∗
⇐⇒ v, T w = 0 for all v ∈ V
⇐⇒ Tv, w = 0 for all v ∈ V
⇐⇒ w ∈ (range T) .
⊥
Thus null T ∗ = (range T) , proving (a).
⊥
If we take the orthogonal complement of both sides of (a), we get (d),
where we have used 6.33. Finally, replacing T with T in (a) and (d) gives
∗
(c) and (b).

If F = R, then the The conjugate transpose of an m-by-n matrix is the n-by-m matrix
conjugate transpose of obtained by interchanging the rows and columns and then taking the
a matrix is the same as complex conjugate of each entry. For example, the conjugate transpose
its transpose, which is of

the matrix obtained by 2 3 + 4i 7
interchanging the rows 6 5 8i
and columns. is the matrix
 
2 6
 3 − 4i 5 

 .
7 −8i
The next proposition shows how to compute the matrix of T ∗ from
the matrix of T. Caution: the proposition below applies only when