Prove that Hv?^T *Hv = I, and conclude that the Householder matrix Hv is unitary. (v^*v...
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Prove that Hv?^T *Hv = I, and conclude that the Householder
matrix Hv is unitary. (v^*v...
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representati...
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representation with respect to every basis of V. Prove that the dimension of linear transform ation that has the same that T must be a scalar multiple of the identity transformation. You can assume V is 3
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T :...
Let T be a linear operator on F2. Prove that if v f 0 is not...
Let T be a linear operator on F2. Prove that if v f 0 is not an eigenvector for T, then v is a cyclic vector for T. Conclude that either T has a cyclic vector T is a scalar multiple of the identity.
Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1)...
Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1)
Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1)
27. Prove that the determinant of the matrix 2 Y3 -I is 2, where (y)(y2()(ys)2. Prove also that the inverse of the matrix G is G(G-I)T İs an orthogonal matrix. Show also that the vector Show that the...
27. Prove that the determinant of the matrix 2 Y3 -I is 2, where (y)(y2()(ys)2. Prove also that the inverse of the matrix G is G(G-I)T İs an orthogonal matrix. Show also that the vector Show that the matrix A is an eigenvector for the matrix A and determine the corresponding eigenvalue
27. Prove that the determinant of the matrix 2 Y3 -I is 2, where (y)(y2()(ys)2. Prove also that the inverse of the matrix G is G(G-I)T İs an...
Let u(t) =t^3 i + ln(t) j + e^2t k and v(t) = 1/t^3 i +...
Let u(t) =t^3 i + ln(t) j + e^2t k and v(t) = 1/t^3 i + 2 j +
t k
2. Let u(t) - ti+In(t)j+ et k and Compute the derivative of the dot product f u(t)v() in two ways and confirm they agree: Compute the dot product u(t) v(t) first and then differentiate the result. . Alternatively, use the following "Dot Product Rule" u(t) v(t)] u'(t) v(t)+ u(t) v'(t) Aside: It's worth noting that there are other forms...
1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b...
1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b) λ is an eigenvalue of T iff λ is an eigenvalue of B. Consequently, A and B have the same eigenvalues (c) There exists an invertible matrix V such that A-V-BV
1. Let V be a vector space with bases B...
2. Work with matrix representations of linear transformations and use knowledge of matrix properties to prove...
2. Work with matrix representations of linear transformations and use knowledge of matrix properties to prove that if a EC is an eigenvalue of a linear operator T:V + V on a (finite-dimensional) inner product space V over C, then ā is an eigenvalue of the adjoint operator T* :V + V. Hint: Check that det (Tij) = det (ij) and utilize this property.
e:= 3. (i) Let T = (V, E) be a graph. Prove that the following are...
e:= 3. (i) Let T = (V, E) be a graph. Prove that the following are equivalent: (a) T is maximally acyclic: T does not contain a cycle but, for any u tv in V with {u, v} not in E, the graph (V, E U{e}) does contain a cycle. (b) T is minimally connected: T is connected but, for any e E E, the graph (V, E {e}) is not connected. (ii) Suppose that I (V, E) is a...
3. Prove valid by a deductive proof: 1. S (TR) 2. R R 3. (V S)-(W...
3. Prove valid by a deductive proof: 1. S (TR) 2. R R 3. (V S)-(W T)/ .. V D~W 4. Prove valid by a deductive proof: 1. (B. L)VT 2. (BVC) (~LO M) 3.~M /.. T 5. Prove valid by a deductive proof: 1. E.(FVG) 2. (E.G)(HVI) 3. (~HV I)(E . F) /.. H= I
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transfor...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representation with respect to every basis of V. Prove that the dimension of linear transform ation that has the same that T must be a scalar multiple of the identity transformation. You can assume V is 3
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T :...
Let T be a linear operator on F2. Prove that if v f 0 is not an eigenvector for T, then v is a cyclic vector for T. Conclude that either T has a cyclic vector T is a scalar multiple of the identity.
Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1)
Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1)
27. Prove that the determinant of the matrix 2 Y3 -I is 2, where (y)(y2()(ys)2. Prove also that the inverse of the matrix G is G(G-I)T İs an orthogonal matrix. Show also that the vector Show that the matrix A is an eigenvector for the matrix A and determine the corresponding eigenvalue
27. Prove that the determinant of the matrix 2 Y3 -I is 2, where (y)(y2()(ys)2. Prove also that the inverse of the matrix G is G(G-I)T İs an...
Let u(t) =t^3 i + ln(t) j + e^2t k and v(t) = 1/t^3 i + 2 j +
t k
2. Let u(t) - ti+In(t)j+ et k and Compute the derivative of the dot product f u(t)v() in two ways and confirm they agree: Compute the dot product u(t) v(t) first and then differentiate the result. . Alternatively, use the following "Dot Product Rule" u(t) v(t)] u'(t) v(t)+ u(t) v'(t) Aside: It's worth noting that there are other forms...
1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b) λ is an eigenvalue of T iff λ is an eigenvalue of B. Consequently, A and B have the same eigenvalues (c) There exists an invertible matrix V such that A-V-BV
1. Let V be a vector space with bases B...
2. Work with matrix representations of linear transformations and use knowledge of matrix properties to prove that if a EC is an eigenvalue of a linear operator T:V + V on a (finite-dimensional) inner product space V over C, then ā is an eigenvalue of the adjoint operator T* :V + V. Hint: Check that det (Tij) = det (ij) and utilize this property.
e:= 3. (i) Let T = (V, E) be a graph. Prove that the following are equivalent: (a) T is maximally acyclic: T does not contain a cycle but, for any u tv in V with {u, v} not in E, the graph (V, E U{e}) does contain a cycle. (b) T is minimally connected: T is connected but, for any e E E, the graph (V, E {e}) is not connected. (ii) Suppose that I (V, E) is a...
3. Prove valid by a deductive proof: 1. S (TR) 2. R R 3. (V S)-(W T)/ .. V D~W 4. Prove valid by a deductive proof: 1. (B. L)VT 2. (BVC) (~LO M) 3.~M /.. T 5. Prove valid by a deductive proof: 1. E.(FVG) 2. (E.G)(HVI) 3. (~HV I)(E . F) /.. H= I
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
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