Download GeoLing10:Linear independence. C ¯ ontents: • Linear dependence

Geometria Lingotto.
GeoLing10:Linear independence.
Contents:
¯
• Linear dependence and linear independence.
• Basis, dimension and coordinates.
Recommended lecture: Leling 7.
¯
EXERCISES
1. Tell whether the following vectors are LI:
(a) (1, 1), (2, 2),
(b) (1, 1, 1), (1, 1, 0), (1, 0, 0),
(c) (1, 1), (1, 2), (3, −5),
(d) (1, 1, 1), (1, 1, 0), (1, 0, 0), (2, −2, 2),
2. Are the rows of


1 2 0 1 1 0 9
A =  3 6 0 3 3 0 27  LI??
1 1 0 0 0 0 4
3. Are the columns of


1 2 7 1 1 0 9
A =  −1 6 0 3 3 0 27  LI??
1 1 0 0 0 0 4
Ingegneria dell’Autoveicolo, GL10
1
Geometria
Geometria Lingotto.
4. True or false:
−
−
−
−
−
−
(a) Three vectors →
v1 , →
v2 , →
v3 are LI iff dim(L(→
v1 , →
v2 , →
v3 )) = 3.
−
−
−
−
(b) If the dot product →
v ·→
w = 0, →
v ,→
w are LI.
→
−
→
−
→
−
−
(c) If v , w are LI, their dot product v · →
w = 0.
−
−
−
−
−
(d) if →
v is such that, for any →
w , the vectors →
v ,→
w are LD, then →
v = 0.
5. Prove B := ((1, 1, 1), (1, 1, 0), (1, 0, 0)) is a basis for R3 . Compute the components
in this basis of:
(a) v = (1, 2, 3), (b) v = (1, 1, 1), (c) v = (3, 2, 1), (d) v = (0, 0, 1).
6. (a) Find a basis for the row space of


1 2 0 1 1 0 9
A =  3 6 0 3 3 0 27 
1 1 0 0 0 0 4
(b) Does the row
2 0 1 1 1 0 0
r=
belong to the row space of A? If yes, compute the components of r in the basis of
part (a)
7. (a) Find a basis for the column space of


1 2 0 1 1 0 9
A =  3 6 0 3 3 0 27 
1 1 0 0 0 0 4
(b) Does the column


2
c= 3 
0
belong in the column space of A? If yes, compute its components in the basis given
in (a)
Ingegneria dell’Autoveicolo, GL10
2
Geometria
Geometria Lingotto.
8. Determine a basis and compute the dimension of the following subspaces:
(a) W =

 −5x1 + 2x2 + x3 = 0
x2 + x3 = 0
(b) W =

−4x3 = 0
2x1 − x2 + x3 = 0
3x2 + 4x3 = 0
(c) W = L((2, −2, 1, 0), (0, 3, 4, 0), (3, 4, 5, 0))
(d) W = L((0, 2, −2, 1, 0), (0, 0, 3, 4, 0), (0, 3, 4, 5, 0))
9. Find a basis of R3 that contains (1, 1, 1) and (1, −1, 1).
10. Write a basis of R5 containing (0, 0, 1, 1, 0) and (0, 1, 1, 0, 0).
11. Find the value of k such that the row space of the matrix A has dimension 1,
i.e. rango(A) = 1:
A=
1 2
3 k
12. Find the value of k such that the row space of the matrix A has dimension 1,
i.e. rango(A) = 1:
A=
k 1
1 k
13. Show dim(Mn,m ) = nm.
14. Compute the dimensions of S = {A ∈ Mn×n : At = A} e A = {A ∈ Mn×n : At =
−A}.
P
i j
15. Let Pd (X, Y ) := {P (X, Y ) =
: aij ∈ R} the vector space of
i+j≤d aij X Y
polynomials in X, Y of degree ≤ d. Compute dim(Pd ).
Ingegneria dell’Autoveicolo, GL10
3
Geometria
Geometria Lingotto.
16. A conic C is by definition the set C = {(x, y) : f (x, y) = 0} where f ∈ P2 (X, Y )\P1 (X, Y ),
hence a plane curve defined by a quadratic polynomial. Prove that if p1 , p2 , p3 , p4 , p5
are points on the plane R2 there exists (at least) one conic C such that p1 , p2 , p3 , p4 , p5 ∈
C . Is C unique?
17. Are the following matrices LI?
1 1
1 0
0 1
(a)
,
,
1 1
0 1
1 0
1 −1
1 0
0 1
(b)
,
,
1 1
0 1
1 0
2 1
1 0
1 0
(b)
,
,
1 1
0 1
1 0
Ingegneria dell’Autoveicolo, GL10
4
Geometria