![Let ū= -=[] -- [14]e a?and A=[23] Show (ū, v)=ūTAV=2u1V1 -201V2–2u2v1 +5u2 V2 is an inner product of R](http://img.homeworklib.com/questions/4cd0b430-7496-11ec-8d7f-99cb14800260.png?x-oss-process=image/resize,w_560)
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Let ū= -=[] -- [14]e a?and A=[23] Show (ū, v)=ūTAV=2u1V1 -201V2–2u2v1 +5u2 V2 is an inner...
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.
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u,...
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u, v e R". Show that ul if and only if ||ü + 해2 (c) Let W be a subspace of R" with an orthogonal basis {w1, ..., w,} and let {ö1, ..., ūg} 22 orthogonal basis for W- (i) Explain why{w1, ..., üp, T1, .., T,} is an (ii Explain why the set in (i) spans R". (iii Show that dim(W) + dim(W1) be...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear f...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
Prob 3. Let T E L(V). Show that (v, u)T :=(Tu, u〉 is an inner product...
Prob 3. Let T E L(V). Show that (v, u)T :=(Tu, u〉 is an inner product on V if and only if T is positive (per our definition of positivity.
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of...
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of R'. Definition: (Modified from our book from page 204.) Let V be a subspace of R". Then the set of vectors (61, 72, ..., 5x} is a basis for V if the following two conditions hold. (a) span{61, 62,...,x} = V (b) {61, 62, ..., 5x} is linearly independent. Definition: Standard Basis for R" The the set of vectors {ēi, 72, ..., en) is...
17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R sat...
17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R satisfying the following propertics The map g is bilinear. That is, for all v, v1, V2, w, w1, W2 CV and all t1,2 ER The map g is symmetric. That is, g(v, w) g(w, v) for all v, weV. The map g is positive definite. That is, g(v,v) 0 for a v e V with equality...
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||.
Let V be R, with thestandard inner product. If is a unitary operator on V, show...
Let V be R, with thestandard inner product. If is a unitary operator on V, show that the matrix of U in the standard ordered basis is either cos θ -sin θ sin θ cos θ cos θ sin θ for some real θ, 0-θ < 2T. Let Us be the linear operator corresponding to the first matrix, i.e., Ue is rotation through the angle . Now convince yourself that every unitary operator on V is either a rotation, or...
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈...
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space ...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(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.
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u, v e R". Show that ul if and only if ||ü + 해2 (c) Let W be a subspace of R" with an orthogonal basis {w1, ..., w,} and let {ö1, ..., ūg} 22 orthogonal basis for W- (i) Explain why{w1, ..., üp, T1, .., T,} is an (ii Explain why the set in (i) spans R". (iii Show that dim(W) + dim(W1) be...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
Prob 3. Let T E L(V). Show that (v, u)T :=(Tu, u〉 is an inner product on V if and only if T is positive (per our definition of positivity.
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of R'. Definition: (Modified from our book from page 204.) Let V be a subspace of R". Then the set of vectors (61, 72, ..., 5x} is a basis for V if the following two conditions hold. (a) span{61, 62,...,x} = V (b) {61, 62, ..., 5x} is linearly independent. Definition: Standard Basis for R" The the set of vectors {ēi, 72, ..., en) is...
17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R satisfying the following propertics The map g is bilinear. That is, for all v, v1, V2, w, w1, W2 CV and all t1,2 ER The map g is symmetric. That is, g(v, w) g(w, v) for all v, weV. The map g is positive definite. That is, g(v,v) 0 for a v e V with equality...
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||.
Let V be R, with thestandard inner product. If is a unitary operator on V, show that the matrix of U in the standard ordered basis is either cos θ -sin θ sin θ cos θ cos θ sin θ for some real θ, 0-θ < 2T. Let Us be the linear operator corresponding to the first matrix, i.e., Ue is rotation through the angle . Now convince yourself that every unitary operator on V is either a rotation, or...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
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