A quadratic n x n matrix A is called ortogonal if A ^T ·A = I, where A^T denotes the transposed matrix to A, and I is t

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A quadratic n x n matrix A is called ortogonal if A ^T ·A
= I, where A^T denotes the transposed matrix to A, and I is
the Identity matrix.
Since the determinant is multiplicative and det(A^T) = det(A),
it follows that for an ortogonal matrix A, det(A) = ±1.
An affine transformation f : R^2 → R^2 is given by
f(v) = Av + b is called an isometry if A is an ortogonal
matrix.
A)
let f be an isometry in the plane. Show that f keeps/conserve
distances |v−w| = | f(v)− f(w)| for all v, w in R^2(it can be
useful to apply the matrix-form for scalar products, v ·w =
(v^T)w)

A Quadratic N X N Matrix A Is Called Ortogonal If A T A I Where A T Denotes The Transposed Matrix To A And I Is T 1 (45.85 KiB) Viewed 21 times

Oppgave 3. En kvadratisk nxn-matrise A kalles ortogonal dersom A¹ ·A=I hvor A betegner den transponerte matrisen til A, og / er identitetsmatrisen. Siden determinanten er multiplikativ og det (A¹) = det (A), følger det at for en ortogonal matrise A, så er det (A) = ±1. En affin avbildning f: R² R² gitt ved f(v) = Av+b kalles en isometri eller en stiv bevegelse dersom A er en ortogonal matrise. 2 (a) La f være en isometri i planet. Vis at f bevarer avstander, dvs. |v-w=f(v)-f(w)| for alle v, w € R². (Det kan være nyttig å bruke matrise-formen for skalarproduktet, v-w = vw).