Problem 6.5. Given any vector space E, fof id linear map f: E -» E is an involution if (a) Prove that an involution f is invertible. What is its inverse? (b) Let E1 and E 1 be the subspaces of E defined as follows E1 {u E \ f(u) = u} E_1 {u € E | f (u) = -u}. Prove that we have a direct sum E E1 E-1 Hint. For every u E E, write и+ f(u) u -f(u) 2 2 (c) If E is finite-dimensional and f is an involution, prove that there is some basis of E with respect to which the matrix of f is of the form 0 0 -In-k where Iis the k x k identity matrix (similarly for In-k) and k = dim(E1). Can you give geometric interpretation of the action of f (especially when k n - 1)?
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Escalate
Trending now
This is a popular solution!
Step by step
Solved in 7 steps with 6 images
