Please comment if you need any clarification.
If you find my answer useful please put thumbs up. Thank you.
1. For each of the following inner product spaces V and linear transformations g, find a...
1. For each of the following inner product spaces V and linear transformations g, find a value of y € V for which g(x) = (x, y), for all 1 € V. (i) V=P2(R) with f(t)g(t) dt and g: V + R defined by g(f) = f'(0) + 2f (1). (ii) V = M2x2(C) with the Frobenius inner product, and g:V + C defined by i i g(A) = tr (( 1 1 1
Part 2 please !
1. For each of the following inner product spaces V and linear transformations g, find a value of y € V for which g(x) = (x,y), for all 2 € V. (i) V = P2(R) with f(t)g(t) dt and g: V + R defined by g(f) = f'(0) + f(1). (ii) V = M2x2(C) with the Frobenius inner product, and g: VC defined by 1 g(A) = tr
I need some help, thank you in advance.
1. For each of the following inner product spaces V and linear transformations g, find a value of ye V for which g(x) = (1,y), for all z e V. (i) V = P2 (R) with (8,9) = S slog(t) dt and g: VR defined by g(f) = f'(0) + 2f(1). (ii) V = M2x2(C) with the Frobenius inner product, and g :V → C defined by g(A) = tr 1+ 1...
3. Consider the inner product space V = M2x2(C) with the Frobenius inner product, and let T:V + V be the linear operator defined by T(A) -1 (i) Compute T:((1+i :)) (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
3. Consider the inner product space V = M2x2(C) with the Frobenius inner product, and let T:V → V be the linear operator defined by 0 T(1) = ( ; ;) A. (i) Compute To((.) (ii) Determine whether or not there is an orthonormal basis of eigenvectors B for which [T]is diagonal. If such a basis exists, find one.
2. Consider the inner product space V = P2(R) with (5,9) = . - f(t)g(t) dt, and let T:V + V be the linear operator defined by T(F) = xf'(x) + 2f (x). (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]2 is diagonal. If such a basis exists, find one.
Consider the inner product space V = P2(R) with (5,9) = { $(0)g(t) dt, and let T:VV be the linear operator defined by T(f) = x f'(x) +2f (x) +1. (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]k is diagonal. If such a basis exists, find one.
Thank you in advance
its asking to find a value y in V for which g(x) =
<x,y> , for all x in V
(*) V = M2x2(C) with the Frobenius inner product, and g: VC defined by 2 g(A) = tr (('+--))
Part 2 please !!
3. Consider the inner product space V = M2x2(C) with the Frobenius inner product, and let T:V + V be the linear operator defined by T(4) = ( ; A. (i) Compute "((1+i (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which (T), is diagonal. If such a basis exists, find one.
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...