Final Week Worksheet
Renzo’s math 469
This worksheet constitutes the in class/take home part of your final exam.
You are encouraged to work collaboratively, and to ask questions to Jess and
Rachel. In the end, please write up your own excellent, tidy, beautifully
presented and eloquent solutions for 6 of the problems and turn them in to
me electronically or physically by Tuesday 5/10.
The determinant
Recall that given a vector space V with a basis e1 , . . . en , the k-th alternating
product Λk (V ) is a vector space of dimension nk with basis given by all
symbols ei1 ∧ . . . ∧ eik with i1 < i2 < . . . , ik .
If T : V → W is a linear function, then Λk (T ) : Λk (V ) → Λk (W ) is
defined on decomposable tensors by:
Λk (T )(v1 ∧ . . . ∧ vn ) = T (v1 ) ∧ . . . ∧ T (vn )
and extended to arbitrary linear combinations of tensors by linearity.
Problem 1. Let V be a four dimensional vector space with bases e1 , e2 , e3 , e4 ,
and W a three dimensional vector space with basis f1 , f2 , f3 . Let T : V → W
be the linear transformation defined by:
T (e1 ) = f1 + 2f3
T (e2 ) = 2f2 + f3
T (e3 ) = 3f1 + 4f2
T (e4 ) = f1 + f3
Write a matrix representing the linear transformation
Λ2 (T ) : Λ2 (V ) → Λ2 (W ).
Recall the axioms of a functor.
A covariant functor from the category of vector spaces to itself is an
assignment F such that:
1. For any vector space V , F(V ) is a vector space.
2. For any linear map T : V → W , F(T ) is a linear map F(T ) : F(V ) →
F(W ).
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3. For any V ,
F(IdV ) = IdF(V )
4. For any pair of linear functions:
T
S
U → V → W,
we have that
F(S ◦ T ) = F(S) ◦ F(T )
Problem 2. Prove that for any positive integer k, the k-th alternating product Λk is a covariant functor from the category of vector spaces to the category of vector spaces.
Problem 3. If V is a k-vector space of dimension n, prove that Λn (V ) ∼
= k,
k
and Λ (V ) = 0 for k > n.
Problem 4. Let T ∈ L(R2 ) be represented by the matrix
a b
M (T ) =
c d
Describe the linear transformation Λ2 (T ).
Definition 1. Let V be a vector space of dimension n and T ∈ L(V ). Then
Λn (T ) : Λn (V ) ∼
= k → Λn (V ) ∼
= k must be multiplication by a scalar factor,
which we define to be the determinant of T .
Problem 5. Compute the determinant of the following linear transformations:
• T defined by T (e1 ) = e1 + 2e2 , T (e2 ) = 3e1 + e3 , T (e3 ) = 5e2 − e3 ;
• F defined by F (x, y, z) = (x + 3z, 2x − y, y + 7z);
• U represented by the matrix


2 1 0
M (U ) =  0 1 −2 
3 0 3
Problem 6. Show that the determinant of the identity function Id. : V → V
is equal to 1.
Problem 7. Prove that the determinant of a composition of operators equals
the product of determinants. If S, T ∈ L(V ):
det(S ◦ T ) = det(S)det(T )
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Problem 8. Prove that if V is an n-dimensional vector space and T ∈
L(V ), λ ∈ k:
det(λT ) = λn det(T )
Problem 9. Show that in general
(v1 +w1 )∧(v2 +w2 )∧. . .∧(vn +wn ) 6= (v1 ∧v2 ∧. . .∧vn )+(w1 ∧w2 ∧. . .∧wn )
and explain how this is the reason for the determinant not being additive:
det(T + S) 6= det(T ) + det(S)
Problem 10. If the matrices A1 and A2 are related by interchanging two
columns, show that
det(A1 ) = −det(A2 )
Problem 11. Show that if a column in a matrix A is a linear combination
of the other columns then
det(A) = 0
Problem 12. For T ∈ L(V ), show that
det(T ) = 0 ⇐⇒ T is not invertible.
Problem 13. Let A = [aji ] be an upper (or lower) triangular matrix. Prove
that the determinant of A is the product of the diagonal entries
Y
det(A) =
aii
i
Problem 14. Let V be a complex vector space over C, and T ∈ L(V ).
Prove that ϕ(λ) = det(T − λId.) is the characteristic polynomal of T . Show
that det(T ) is the constant coefficient of the characteristic polynomial.
Problem 15. Given a matrix A, prove that
det(A) = det(AT )
as follows. Assume that A represents a linear operator T on V a complex
vector space (You can assume that the transformation is on a complex vector
space even if A has real entries). Then you can choose a basis such that...
Problem 16. Let V be a real vector space with an inner product. Show that
an isometry S ∈ L(V ) has determinant ±1
Problem 17. Consider R2 with the standard inner product, and standard
orthonormal basis e1 , e2 . Show that for any pair of vectors v, w,
v ∧ w = α(e1 ∧ e2 )
where |α| is the area of the parallelogram determined by v and w. Prove (I
suggest by induction) that this statement generalizes to arbitrary dimension,
by replacing parallelogram with n-dimensional parallelepiped and area by ndimensional volume.
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Problem 18. Let V = Rn with the standard inner product, and T ∈ L(V ).
The number det(T ) is the constant of proportionality by which the volume
of any shape in V is scaled by T . Don’t show this in full generality though
(to do so requires integration and all that jazz, and we are not focusing on
that), but prove it for parallelepipeds of the form v1 ∧ . . . ∧ vn .
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