MA20008 Algebra 1, 2004, Sheet 8

Geoff Smith, http://www.bath.ac.uk/∼masgcs

1. Suppose that V is a finite dimensional vector space over a field F, and that α : V ⟶ V is a linear map. Suppose that e_{1}, e_{2}, …, e_{n} is a basis of V such that each subspace ⟨e_{i}⟩ is α-invariant. Prove that the matrix of α with respect to this basis (in both domain and codomain) is diagonal.

2. We consider the ordinary inner product on ℝ^{2} or ℝ^{3} defined as the “dot” or “scalar” product. Let the vertices of triangle ABC have position vectors a, b and c respectively.

(a) Show that the point G with position vector \frac{1}{3}(a + b + c) lies on each median (a line joining a vertex to the midpoint of the opposite side). Conclude that the medians are concurrent (at a point G which is called the centroid of △ABC).

(b) The altitude of △ABC through A is the straight line through A which is perpendicular to BC. Show that a point P is on this altitude if and only if the position vector r of P satisfies (r − a) · (b − c) = 0.

(c) Now we insist that the origin is the circumcentre O of △ABC. Thus there is a quantity R such that

a · a = b · b = c · c = R^{2}.

Show that the point H with position vector a + b + c is on all three altitudes. Deduce that the three altitudes of △ABC are concurrent at H. The point H is called the orthocentre of △ABC.

(d) Deduce that the three points O, G and H are colinear, and that the distances are such that |OH| = 3|OG|. The line through O and H is called the Euler line of △ABC.

(e) Let L be the midpoint of BC and M be the midpoint of AH. Let N be the midpoint of LM. Find the position vector of N (with the origin still at O).

(f) Deduce that N is the midpoint of OH so that O, G, N, H are colinear and the ratios of lengths are

|OG| : |GN| : |NH| = 2 : 1 : 3.

(g) Show that |LM| = R.

(h) Deduce that the circle with centre N and radius R/2 goes through the following nine interesting points: the midpoints of the sides of △ABC, the feet of the altitudes of △ABC and the three points which are midway between H and each of the three vertices A, B and C. This is the ‘nine-point circle’ or ‘Feuerbach circle’.

(a) Let ABCD be a cyclic quadrilateral. A maltitude is a straight line through the midpoint of a side which perpendicular to the opposite side. Show that the four maltitudes are concurrent.

3. Suppose that V is an inner product space, with inner product denoted by ⟨ , ⟩. Suppose that U and W are subspaces of V.

(a) Show that (U + W)^{⊥} = (U ∪ W)^{⊥} = U^{⊥} ∩ W^{⊥}.

(b) Suppose that U ≤ W. Show that W^{⊥} ≤ U^{⊥}.

4. Suppose that V is an inner product space of dimension n, and 0 ≠ v ∈ V. Prove that dim({v}^{⊥}) = n − 1.