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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.
See the step by step solution

## Step 1: Example 1: A quadratic function

Consider the following quadratic function: $$f(x) = x^2$$. To show that it is non-linear, we will test conditions (i) and (ii). Let's test condition (i): $$f(x_1 + x_2) = (x_1 + x_2)^2 = x_1^2 + 2x_1x_2 + x_2^2$$ $$f(x_1) + f(x_2) = x_1^2 + x_2^2$$ Since $$f(x_1 + x_2) \neq f(x_1) + f(x_2)$$, condition (i) is not satisfied. Thus, the quadratic function $$f(x) = x^2$$ is not a linear transformation.

## Step 2: Example 2: An exponential function

Consider the exponential function: $$g(x) = e^x$$. To show that it is non-linear, we will test conditions (i) and (ii). Let's test condition (i): $$g(x_1 + x_2) = e^{x_1 + x_2}$$ $$g(x_1) \cdot g(x_2) = e^{x_1} \cdot e^{x_2}$$ Since $$g(x_1 + x_2) = g(x_1) \cdot g(x_2)$$, condition (i) is not satisfied. Thus, the exponential function $$g(x) = e^x$$ is not a linear transformation.

## Step 3: Example 3: An absolute value function

Consider the absolute value function: $$h(x) = |x|$$. To show that it is non-linear, we will test conditions (i) and (ii). Let's test condition (i): $$h(x_1 + x_2) = |x_1 + x_2|$$ $$h(x_1) + h(x_2) = |x_1| + |x_2|$$ Since $$h(x_1 + x_2) \neq h(x_1) + h(x_2)$$ in general (consider negative values), condition (i) is not satisfied. Thus, the absolute value function $$h(x) = |x|$$ is not a linear transformation.

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