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Problem 52

A linear transformation, \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\), is a function defined on a vector space \(\mathrm{V}\) over a field \(\mathrm{K}\) that satisfies i) \(\mathrm{T}\left(\mathrm{v}_{1}+\mathrm{v}_{2}\right)=\mathrm{T}\left(\mathrm{v}_{1}\right)+\mathrm{T}\left(\mathrm{v}_{2}\right)\) ii) $\mathrm{T}\left(\alpha \mathrm{v}_{1}\right)=\alpha \mathrm{T}\left(\mathrm{v}_{1}\right)$ for \(\mathrm{v}_{1}, \mathrm{v}_{2} \varepsilon \mathrm{V}\) and $\alpha \varepsilon \mathrm{K}$. Give examples of some non-linear functions by showing that they fail to satisfy either (i) or (ii).

Expert verified

Here are three examples of non-linear functions:
1. The quadratic function \(f(x) = x^2\) fails to satisfy condition (i), as \(f(x_1+x_2) \neq f(x_1)+f(x_2)\).
2. The exponential function \(g(x) = e^x\) fails to satisfy condition (i), as \(g(x_1+x_2) \neq g(x_1)g(x_2)\).
3. The absolute value function \(h(x) = |x|\) fails to satisfy condition (i), as \(h(x_1+x_2) \neq h(x_1)+h(x_2)\) in general for negative values.

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Chapter 2

Find the inverses of the following transformations using the matrices associated with the transformations: 1) \(\mathrm{T}\) is a counterclockwise rotation in \(\mathrm{R}^{2}\) through the angle \(\theta\). 2) \(\mathrm{T}\) is reflection in the \(\mathrm{y}\) -axis in \(\mathrm{R}^{2}\)

Chapter 2

1 ) Let \(\mathrm{V}\) be a vector space over a field \(\mathrm{F}\) and let \(\alpha \varepsilon \mathrm{F}\) be any nonzero scalar. Show that multiplication by \(\alpha\), $\mathrm{M}_{\alpha}: \mathrm{V} \rightarrow \mathrm{V}$ is a vector-space isomorphism. 2) Let \(\mathrm{V}\) and \(\mathrm{w}\) be any vector spaces over \(\mathrm{F}\) and \(\mathrm{V} \varepsilon \mathrm{W}\) their Cartesian product. Show that \(\mathrm{V} \cong \mathrm{V} \times\\{0\\}\) where \(0 \varepsilon \mathrm{W}\).

Chapter 2

Let T: \(\mathrm{R}^{4} \rightarrow \mathrm{R}^{3}\) be a linear transformation defined by $\mathrm{T}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t})=(\mathrm{x}-\mathrm{y}+\mathrm{z}+\mathrm{t}, \mathrm{x}+2 \mathrm{z}-\mathrm{t}, \mathrm{x}+\mathrm{y}+3 \mathrm{z}-3 \mathrm{t})$ Find a basis and the dimension of the i) image of \(\mathrm{T}\) ii) kernel of \(\mathrm{T}\).

Chapter 2

Let the transformation \(L: R^{3} \rightarrow R^{3}\) be defined by $L([x, y, z])=[x, y]$ Show that \(\mathrm{L}\) is a linear transformation and describe its effect.

Chapter 2

Find the inverse transformation of the following linear transformation : i) $\mathrm{T}(\mathrm{x}, \mathrm{y})=(2 \mathrm{x}+\mathrm{y},-\mathrm{x}+3 \mathrm{y})$ ii) \(T(x, y, z)=(x+y+z, x-y+z,-x+y+z)\)

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