Preserve every linear combination
A map is linear when T(av+bw)=aT(v)+bT(w) for all vectors and scalars in the chosen field. In particular, a linear map sends zero to zero, though that condition alone is insufficient.
Algebra / English
Linear Transformation (Linear Mapping) in Linear Algebra
Let's Get Complex · YouTube · 8:37
YouTube · Let's Get ComplexOriginal video
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Recognize linear transformations through preserved combinations and examples including rotations, projections and differentiation.
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Chapters
0:00Preserve every linear combination3:00Compose linear geometric actions6:00Vectors can be functionsLearning script
Generated from the video's visuals and explanation; not verbatim speech.
A map is linear when T(av+bw)=aT(v)+bT(w) for all vectors and scalars in the chosen field. In particular, a linear map sends zero to zero, though that condition alone is insufficient.
Rotation about the origin and uniform scaling both preserve vector addition and scalar multiplication. Their composition, such as rotating 30 degrees and scaling by two, remains linear.
Projection onto the horizontal axis is a linear map. Differentiation on a polynomial space is also linear because the derivative of a linear combination is the same combination of derivatives.
Knowledge cards
Compose linear geometric actions
Rotation about the origin and uniform scaling both preserve vector addition and scalar multiplication. Their composition, such as rotating 30 degrees and scaling by two, remains linear.
Vectors can be functions
Projection onto the horizontal axis is a linear map. Differentiation on a polynomial space is also linear because the derivative of a linear combination is the same combination of derivatives.
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- Linear transformations ExplanationAt 0:00
Why this connection?
A map is linear when T(av+bw)=aT(v)+bT(w) for all vectors and scalars in the chosen field. In particular, a linear map sends zero to zero, though that condition alone is insufficient. Rotation about the origin and uniform scaling both preserve vector addition and scalar multiplication. Their composition, such as rotating 30 degrees and scaling by two, remains linear. Projection onto the horizontal axis is a linear map. Differentiation on a polynomial space is also linear because the derivative of a linear combination is the same combination of derivatives.
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