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Problem 563

Let \(\mathrm{V}\) be the vector space of all polynomial functions from \(\mathrm{R}\) into \(\mathrm{R}\) with degree less than or equal to \(2 .\) Find a basis for \(\mathrm{V}\) by using the following procedures i) Find three linear functionals on \(\mathrm{V}\) ii) Use these functionals as a basis for \(\mathrm{V}^{*}\), the dual of \(\mathrm{V}\). iii) Use the functionals to find a basis for \(\mathrm{V}\).Annthilators, Transposes \& Adjoint

Expert verified

To find a basis for the vector space V of all polynomial functions from R into R with degree less than or equal to 2, we first find three linear functionals \(L_1\), \(L_2\), and \(L_3\) that evaluate polynomials at specific points (0, 1, and -1, respectively). Then, we use these functionals to form a basis for the dual space V* as \(\mathcal{B}^* = \{L_1, L_2, L_3\}\). Finally, we find the annihilators for each functional and solve the resulting system of equations to obtain the basis for V: \(\mathcal{B} = \{1 + x, x^2\}\).

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Chapter 22

Let \(\mathrm{V}\) be the space of polynomials over the field of complex numbers, with the inner product $$ (\mathrm{f} / \mathrm{g})={ }^{1} \int_{0} \mathrm{f}(\mathrm{t}) \mathrm{g}_{-}(\mathrm{t}) \mathrm{dt} $$ Let \(\mathrm{D}\) be the differentiation operator on \(\mathrm{C}[\mathrm{x}]\). Show that \(\mathrm{D}\) has no adjoint.Multilinear Functionals

Chapter 22

Let \(\mathrm{V}\) be the space of polynomials over the field of complex numbers, with the inner product $$ (\mathrm{f} / \mathrm{g})=1 \int_{0} \mathrm{f}(\mathrm{t}) \mathrm{g}(\mathrm{t}) \mathrm{dt} $$ (where the bar indicates complex conjugation). Consider the operator 'multiplication by \(\mathrm{f}^{\prime}\), that is, the linear operator \(\mathrm{M}_{\mathrm{f}}\) defined by \(\mathrm{M}_{\mathrm{f}}(\mathrm{g})=\mathrm{fg}\). What is the adjoint of this operator?

Chapter 22

Let \(\mathrm{V}\) be an \(\mathrm{n}\) -dimensional vector space over the field \(\mathrm{F}\) and let \(\mathrm{T}\) be a linear operator on \(\mathrm{V} .\) Suppose \(\mathrm{B}=\left(\alpha_{1}, \ldots, \alpha_{\mathrm{n}}\right\\}\) and $B^{\prime}=\left\\{\alpha_{1}^{\prime}, \ldots, \alpha_{n}^{\prime}\right\\}\( are two ordered bases for \)\mathrm{V}$. Show how the transpose transformation \(\mathrm{T}^{\mathrm{t}}\) can be used to derive the formula for changing bases, that is, to find \(\mathrm{T}\) in the ordered basis B' given the matrix of \(\mathrm{T}\) in \(\mathrm{B}\).

Chapter 22

Let \(\mathrm{V}\) be \(\mathrm{K}^{\mathrm{n} \times \mathrm{n}}\) with the inner product $(\mathrm{A} / \mathrm{B})=\operatorname{tr}\left(\mathrm{B}^{*} \mathrm{~A}\right)\(. Let \)\mathrm{M}$ be a fixed \(n \times n\) matrix over \(K\) (a field). What is the adjoint of left multiplication by \(\mathrm{M}\) ?

Chapter 22

Let \(W\) be the subspace of \(R^{5}\) which is spanned by the Vectors $$ \begin{aligned} &\alpha_{1}=(2,-2,3,4,-1), \alpha_{2}=(-1,1,2,5,2) \\ &\alpha_{3}=(0,0,-1,-2,3) & \alpha_{4}=(1,-1,2,3,0) \end{aligned} $$ Describe \(\mathrm{W}^{\circ}\), the annihilator of \(\mathrm{W}\).

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