Compatibility of linear operators with scalar multiplication: A linear map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>f</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle f}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> is compatible with scalar multiplication, i.e. from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=\lambda v}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>w</mi><mo>=</mo><mi>λ<!-- λ --></mi><mi>v</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle w=\lambda v}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcce0057291cb5fe79e97340b7dbad6bf950256" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.245ex; height:2.176ex;" alt="{\displaystyle w=\lambda v}"></span> follows <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(w)=\lambda f(v)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ<!-- λ --></mi><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle f(w)=\lambda f(v)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d08291569438a97642454140867239e46656887" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.421ex; height:2.843ex;" alt="{\displaystyle f(w)=\lambda f(v)}"></span>. This property can be condensed in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\lambda v)=\lambda f(v)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>f</mi><mo stretchy="false">(</mo><mi>λ<!-- λ --></mi><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ<!-- λ --></mi><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle f(\lambda v)=\lambda f(v)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9841384b6ea81089c1a3c95daef0f6a1a5b2bef8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.24ex; height:2.843ex;" alt="{\displaystyle f(\lambda v)=\lambda f(v)}"></span>. For a non linear map this property might not hold. There might be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in K,v\in V}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>λ<!-- λ --></mi><mo>∈<!-- ∈ --></mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>∈<!-- ∈ --></mo><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \lambda \in K,v\in V}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc27548aff50e70960b253059f812784664a1c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:13.051ex; height:2.509ex;" alt="{\displaystyle \lambda \in K,v\in V}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\lambda v)\neq \lambda f(v)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>f</mi><mo stretchy="false">(</mo><mi>λ<!-- λ --></mi><mi>v</mi><mo stretchy="false">)</mo><mo>≠<!-- ≠ --></mo><mi>λ<!-- λ --></mi><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle f(\lambda v)\neq \lambda f(v)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d96c49b70ab4ce14df8c2ec472a5e77a46e80a9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.24ex; height:2.843ex;" alt="{\displaystyle f(\lambda v)\neq \lambda f(v)}"></span>.
<a rel="nofollow" class="external text" href="http://kulla.me">Stephan Kulla</a> (<a href="//commons.wikimedia.org/wiki/User:Stephan_Kulla" title="User:Stephan Kulla">User:Stephan Kulla</a>)
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