PROBLEM SET III
DUE FRIDAY,  FEBRUARY
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De nition. Suppose V1 , . . . , Vn vector spaces. We call a function F : ni=1 Vi . R
multilinear if for any 1 ≤ i ≤ n and any vectors vj ∈ Vj (one for each j ̸= i),
the map Vi . R given by
x.
F(v1 , . . . , vi−1 , x, vi+1 , . . . , vn )
is linear. Call F alternating if for any w1 , . . . , wn ∈ V, if wi = wj with i ̸= j,
then
F(w1 , . . . , wn ) = 0.
De nition. An elementary m × m matrix is one of the following:
() for 1 ≤ i < j ≤ m, the matrix
Σij := (e1 | · · · | ei−1 | ej | ei+1 | · · · | ej−1 | ei | ej+1 | · · · | em ),
() for 1 ≤ i ≤ m and α ∈ R − {0}, the matrix
Mi (α) := (e1 | · · · | ei−1 | αei | ei+1 | · · · | em ),
or
() for 1 ≤ i ≤ m, 1 ≤ j ≤ m with i ̸= j, and α ∈ R, the matrix
(
)
Pij (α) := e1 | · · · | ei−1 | ei + αej | ei+1 | · · · | em ,
Exercise⋆ . Prove that the following are equivalent for an m × m matrix M.
(a) e linear transformation Rm . Rm corresponding to M is an isomorphism.
(b) e linear transformation Rm . Rm corresponding to M is injective.
(c) e linear transformation Rm . Rm corresponding to M is surjective.
(d) e columns of M are linearly independent.
(e) e columns of M span Rm .
(f ) e rows of M are linearly independent.
(g) e rows of M span (Rm )∨ .
(h) M can be written as a product of elementary matrices.
(i) For any alternating multilinear map F : (Rm )m . R,
F(Me1 , . . . , Mem ) ̸= 0.
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