Two required properties
A linear map satisfies and T(ca)=cT(a) for all vectors and scalars in its domain. Testing only a few numerical examples cannot prove these universal conditions.
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Linear transformations | Matrix transformations | Linear Algebra | Khan Academy
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Check linear transformations by verifying preservation of addition and scalar multiplication.
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1:30Two required properties4:30A linear example12:00A nonlinear counterexampleLearning script
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A linear map satisfies and T(ca)=cT(a) for all vectors and scalars in its domain. Testing only a few numerical examples cannot prove these universal conditions.
For T(x₁,x₂)=(x₁+x₂,3x₁), componentwise expansion verifies both properties for arbitrary vectors and scalars.
For T(x₁,x₂)=(x₁²,0), scalar multiplication introduces c² rather than c. For example and (x₁,x₂)=(1,0) violate homogeneity, so the map is not linear.
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A linear example
For T(x₁,x₂)=(x₁+x₂,3x₁), componentwise expansion verifies both properties for arbitrary vectors and scalars.
A nonlinear counterexample
For T(x₁,x₂)=(x₁²,0), scalar multiplication introduces c² rather than c. For example and (x₁,x₂)=(1,0) violate homogeneity, so the map is not linear.
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A linear map satisfies and T(ca)=cT(a) for all vectors and scalars in its domain. Testing only a few numerical examples cannot prove these universal conditions. For T(x₁,x₂)=(x₁+x₂,3x₁), componentwise expansion verifies both properties for arbitrary vectors and scalars. For T(x₁,x₂)=(x₁²,0), scalar multiplication introduces c² rather than c. For example and (x₁,x₂)=(1,0) violate homogeneity, so the map is not linear.
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