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![VI, V z EV and Vive are orthogonal, Will till v2 15 = 1 Vital Now = Il vill tvar <1,7 + LV2, 27 = [vi, ve] + [N, V2> + (~2,17](http://img.homeworklib.com/questions/7cfc8c90-10ba-11eb-9392-f73299bd03ce.png?x-oss-process=image/resize,w_560)
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2. Suppose that V is an inner product space. (i) Prove that, for any vectors 01,...
3. Let V be a finite dimensional inner product space, and suppose that T is a...
3. Let V be a finite dimensional inner product space, and suppose that T is a linear operator on this space. (i) Let B be an ordered orthonormal basis for V and let U be the linear operator on V determined by [U19 = (T);. Then, for all 01,09 € V, (01, T(02)) = (U(V1), v2) (ii) Prove that the conclusion of the previous part does not hold, in general, if the basis 8 is not orthonormal.
Suppose V is a finite dimensional inner product space, and dim V = n. If is...
Suppose V is a finite dimensional inner product space, and dim V
= n.
If is an orthogonal subset
of V, prove that
a. W can be extended to an orthogonal basis for V.
b. is an orthogonal basis
for
c.
Question 6) (9 points) Prove each of the following statements. (a) Suppose that the vectors {v,...
Question 6) (9 points) Prove each of the following statements. (a) Suppose that the vectors {v, w, u} are linearly independent vectors in some vector space V. Prove then that the vectors {v + w, w + u,v + u} are also linearly independent in V. (b) Suppose T is a linear transformation, T: P10(R) → M3(R) Prove that T cannot be 1-to-1 (c) Prove that in ANY inner product that if u and w are unit vectors (ie ||vl|...
Let v and w be vectors in an inner product space V. Show that v is...
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||.
1). Let V be an n-dimensional inner product space, let L be a linear transformation L...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
Question 2 Suppose that T :V +V is a linear transformation. Further suppose {01, v2} forms...
Question 2 Suppose that T :V +V is a linear transformation. Further suppose {01, v2} forms a basis for V. Prove that if T(01) = v1 + 09 and T (12) = V1 – 02 then T is both 1-to-1 and onto.
(1 point) Let {uj, u2, u2 ) be an orthonormal basis for an inner product space...
(1 point) Let {uj, u2, u2 ) be an orthonormal basis for an inner product space V. Suppose y = qui + buz + cuz is so that|lvl1 = V116. (v, uz) = 10, and (v. uz) = 4. Find the possible values for a, b, and c. a = CE (1 point) Suppose U1, U2, Uz is an orthogonal set of vectors in Rº. Let w be a vector in Span(v1, 02, 03) such that UjUi = 42, 02.02...
1.(16) Let P be an inner product space with an inner product defined as <.g >...
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...
By justifying your answer, determine whether the function (, ) defines an inner product on V....
By justifying your answer, determine whether the function (, ) defines an inner product on V. (a) ((u1, U2, U3, U4), (V1, V2, 03, 04)) = U104 – 5u2 V3 and V = R4. (b) ((uj, u2), (01, 02)) = V2 U1V1 + u202 and V = R2.
(1 point) Consider the complex inner product space C with the usual inner product Let -4i...
(1 point) Consider the complex inner product space C with the usual inner product Let -4i 4i and let w = span(vi,V2). (a) Compute the following inner products: (v.vi)- 2 (Vi, V2-12 (2. V)12 (b) Apply the Gram-Schmidt procedure to Vi and v2 to find an orthogonal basis (ui,u2l for W , u2=1112
3. Let V be a finite dimensional inner product space, and suppose that T is a linear operator on this space. (i) Let B be an ordered orthonormal basis for V and let U be the linear operator on V determined by [U19 = (T);. Then, for all 01,09 € V, (01, T(02)) = (U(V1), v2) (ii) Prove that the conclusion of the previous part does not hold, in general, if the basis 8 is not orthonormal.
Suppose V is a finite dimensional inner product space, and dim V
= n.
If is an orthogonal subset
of V, prove that
a. W can be extended to an orthogonal basis for V.
b. is an orthogonal basis
for
c.
Question 6) (9 points) Prove each of the following statements. (a) Suppose that the vectors {v, w, u} are linearly independent vectors in some vector space V. Prove then that the vectors {v + w, w + u,v + u} are also linearly independent in V. (b) Suppose T is a linear transformation, T: P10(R) → M3(R) Prove that T cannot be 1-to-1 (c) Prove that in ANY inner product that if u and w are unit vectors (ie ||vl|...
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||.
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
Question 2 Suppose that T :V +V is a linear transformation. Further suppose {01, v2} forms a basis for V. Prove that if T(01) = v1 + 09 and T (12) = V1 – 02 then T is both 1-to-1 and onto.
(1 point) Let {uj, u2, u2 ) be an orthonormal basis for an inner product space V. Suppose y = qui + buz + cuz is so that|lvl1 = V116. (v, uz) = 10, and (v. uz) = 4. Find the possible values for a, b, and c. a = CE (1 point) Suppose U1, U2, Uz is an orthogonal set of vectors in Rº. Let w be a vector in Span(v1, 02, 03) such that UjUi = 42, 02.02...
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...
By justifying your answer, determine whether the function (, ) defines an inner product on V. (a) ((u1, U2, U3, U4), (V1, V2, 03, 04)) = U104 – 5u2 V3 and V = R4. (b) ((uj, u2), (01, 02)) = V2 U1V1 + u202 and V = R2.
(1 point) Consider the complex inner product space C with the usual inner product Let -4i 4i and let w = span(vi,V2). (a) Compute the following inner products: (v.vi)- 2 (Vi, V2-12 (2. V)12 (b) Apply the Gram-Schmidt procedure to Vi and v2 to find an orthogonal basis (ui,u2l for W , u2=1112
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