Let v=(1,-4,12) and w=(3,5,-1) be vectors.
Compute |v| and |w|.
Let v=(1,-4,12) and w=(3,5,-1) be vectors. What is 2v-3w?
41
and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be ved let c be a scalar. Prove the given property. 39. u. v = v.u 40. (cu) v = c(u.v) = u • (cv) 41. (u + v). w = uw + v.w 42. (u - v)•(u + v) = | u |2 - 1 v 12
Let v and w be vectors in an inner product space V. Show that v is orthogonal to w if and only if ||v + w|| = ||v – w||.
3) Let u a) Treating u, v', and w' as vectors, are the inner products u.v', v'.u, and u.w' defined? If yes, compute them. If any of them is not defined, explain why not. b) Treating u, v', and ' as matrices, are the products uv', v'u, and w' defined? If yes, compute them. If any of them is not defined, explain why not.
Let v and w be two vectors whose modules are equal to 3 and 1, respectively the angle formed between them is equal to π / 6. If θ denotes the measure of the angle between v + w and v-w, how much is cos θ worth?
Exercise 4.12. For the pairs of vectors v, w below, compute projwv and proj,w. Also, verify that v and w – projųw are orthogonal and v and v – projwv are orthogonal. 0 (1) v= (0) -- 2 (2) v= W =
5. Let ū and w be vectors in R3. Prove that (ö - w) x (v + 2) = 2(vx w).
Question 1
Question 2
Let u, v, w be three vectors in R4 with the property that 4u - 30+2w = 0. Let A be the 4 x 2 matrix whose columns are u and u (in that order). Find a solution to the equation Ac =W. Let 1 -2 0 3 A=1 -2 2-1 2 -4 1 4 Find a list of vectors whose span is the set of solutions to Ax = 0. 1 1 Enter the list...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator in V . Show that T has an adjoint, and describe T ∗ explicitly
3 - 2 Let u= Note that {u, v, w} is an orthogonal set of vectors and w - -3 4 9 be a vector in subspace W, where W = Span{u, v, w}. Let y= 11 -27 Write y as a linear combination of u, v, and uw, i.e. y = ciu + cqũ + c3W. Answer: y=