Solvability of linear equations. Let be arbitrary. Wish to find such that and is linear. Find conditions on T such that there is a solution to for each and the solution is unique.
Solvability of linear equation .
We need to find an condition on such that for all there exist such that .
Suppose
Suppose be a basis of so they are linearly independent . If are preimage of repectively then is also an linearly independent set in .
As contains a set of n vectors which are linearly independent so .
As be a basis of so any element of can be written as linear combination of elements of so are innearly independent set in W .
Hence the required condition on for which has a solution for all is ,
and .
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If you any doubt please comment .
Solvability of linear equations. Let be arbitrary. Wish to find such that and is linear. Find...
Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We consider the projection of y onto , i.e., the solution of (1) (a) Prove that is a plane, i.e., if , then for any . (b) Prove that z is a solution of (1) if and only if and (2) (c) Find an explicit solution of (1). ( d) Prove the solution found in part (c) is unique. We...
Using FTLM. a) Let . Use linear algebra to prove that there is a polynomial such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM. b) Let be distinct elements of . Let be any elements of . Use linear algebra to prove that there is a such that Hint: consider the map defined by . You can use any facts from algebra about the solution...
Let be an arbitrary mapping satisfying the properties (S1) - (S4) of Theorem (at the end). Beyond that let Show that the following statements apply to all u, v ∈ Rn. The Theorem: For the scalar product, vectors u, v, w∈ Rn : We were unable to transcribe this imageWe were unable to transcribe this imageu_u We were unable to transcribe this image u_u
Let be an arbitrary function and A X. i) Show that A ii) Give an example to show that in general A = . iii) Show that, if is injective, then A = iv) Show that, if X and Y are modules; is a homomorphism of modules and A is a submodule of X such that ker, then we also have A = We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe...
Let T: V V and S: V V and R: V V be three linear operators on V. Suppose we have T S= S R , Then prove ker(S) is an invariant subspace for R . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let . (a) Find the singular value decomposition of A. (b) Find the least squares solution to the linear system We were unable to transcribe this imageWe were unable to transcribe this image
Use the Gauss-Jordan elimination process on the following system of linear equations to find the value of z. a) z = 5 b) z = 0 c) z = 4 d) z = 2 e) z = 3 f) None of the above. Use the Gauss-Jordan elimination process on the following system of linear equations to find the value of x. a) x = -10 b) x = -21 c) x = -11 d) x = 8 e) x =...
Let Y = Xβ + ε be the linear model where X be an n × p matrix with orthonormal columns (columns of X are orthogonal to each other and each column has length 1) Let be the least-squares estimate of β, and let be the ridge regression estimate with tuning parameter λ. Prove that for each j, . Note: The ridge regression estimate is given by: The least squares estimate is given by: We were unable to transcribe this...
Let V be a finite dimensional inner product space, w1,w2V. Let TL(V) and Tv=<v,w1>w2 for all vV. Find all eigenvalues and the corresponding eigenspaces of T. Please provide full solution. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Partial Differential Equations. Let be the upper half of a disk of radius 1. Solve the Dirichlet problem for the Laplace equation: in for -1 < x <1 and y = 0 for We were unable to transcribe this imageu : We were unable to transcribe this imageWe were unable to transcribe this imageu = y We were unable to transcribe this image u : u = y