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Problem 76

Let \(\left\\{1, \mathrm{x}, \mathrm{x}^{2}\right\\}\) be a basis for the vector space of polynomials of degree not exceeding 2 . Let $\mathrm{D}, \mathrm{D}^{2}\( and \)\mathrm{D}^{3}$ denote differentiation operators. Find the matrices of \(\mathrm{D}, \mathrm{D}^{2}\) and \(\mathrm{D}^{3}\) relative to the above basis.

Expert verified

The matrices of the differentiation operators D, D^2, and D^3 relative to the given basis are:
\[D = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 2\\0 & 0 & 0\end{bmatrix}\]
\[D^2 = \begin{bmatrix}0 & 0 & 2\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\]
\[D^3 = \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\]

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Chapter 3

Let \(\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}\) be given by $\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=\left(4 \mathrm{x}_{1}-2 \mathrm{x}_{2}, 2 \mathrm{x}_{1}+\mathrm{x}_{2}\right)$ and let \(\\{(1,1),(-1,0)\\}\) be a basis for \(\mathrm{R}^{2}\). Compute the matrix of \(\mathrm{T}\) in the given basis.

Chapter 3

Let T: \(\mathrm{R}^{3} \rightarrow \mathrm{R}^{2}\) be given by $\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=\left(7 \mathrm{x}_{1}+2 \mathrm{x}_{2}-3 \mathrm{x}_{3}, \mathrm{x}_{2}\right)$ As bases for \(\mathrm{R}^{3}\) and \(\mathrm{R}^{2}\) respectively, let $\mathrm{G}=\left\\{\mathrm{g}_{1}, \mathrm{~g}_{2}, \mathrm{~g}_{3}\right\\}=\\{(1,0,0),(0,1,-1),(0,0,1)\\}$ $\mathrm{H}=\left\\{\mathrm{h}_{1}, \mathrm{~h}_{2}\right\\}=\\{(1,0),(0,-1)\\}$

Chapter 3

Systems of linear transformations and matrices are isomorphic vector spaces. Show by means of an example that the product of two linear transformations can be represented as the product of two matrices with respect to a given basis.

Chapter 3

Let \(\mathrm{S}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}\) and $\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}\( be given by \)\mathrm{S}(\mathrm{x}, \mathrm{y})=$ \((x-2 y, 2 x+7 y), T(x, y)=(3 x+2 y, x-y) .\) Find \((S+T)(x, y)\) and \((3 \mathrm{~T})(\mathrm{x}, \mathrm{y})\), in both functional and matrix form.

Chapter 3

Let \(\left\\{\mathrm{e}_{i}\right\\}=\\{(1,0),(0,1)\\}\) and \(\left\\{\mathrm{f}_{i}\right\\}=\\{(1,1),(-1,0)\\}\) be two bases for \(\mathrm{R}^{2}\). Find the transition matrix from \(\left\\{\mathrm{e}_{\mathrm{i}}\right\\}\) to \(\left\\{\mathrm{f}_{\mathrm{i}}\right.\) ).

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