Independence
Two nonzero vectors are independent when neither is a scalar multiple of the other. Their linear combinations then span a plane through the origin.
Algebra / English
Independence, Basis, and Dimension
MIT OpenCourseWare · YouTube · 13:19
YouTube · MIT OpenCourseWareOriginal video
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A geometric introduction to linear independence, spanning sets, bases and dimension using vectors in three-dimensional space.
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0:33Independence3:00Redundant vectors9:00Basis and dimensionLearning script
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Two nonzero vectors are independent when neither is a scalar multiple of the other. Their linear combinations then span a plane through the origin.
Adding the sum of two existing vectors does not enlarge their span. The new three-vector set is dependent, so it is not a basis for the plane.
A basis both spans the space and is linearly independent. Different bases of the same finite-dimensional space contain the same number of vectors; this number is its dimension.
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Redundant vectors
Adding the sum of two existing vectors does not enlarge their span. The new three-vector set is dependent, so it is not a basis for the plane.
Basis and dimension
A basis both spans the space and is linearly independent. Different bases of the same finite-dimensional space contain the same number of vectors; this number is its dimension.
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- Basis and dimension ExplanationAt 0:33
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Two nonzero vectors are independent when neither is a scalar multiple of the other. Their linear combinations then span a plane through the origin. Adding the sum of two existing vectors does not enlarge their span. The new three-vector set is dependent, so it is not a basis for the plane. A basis both spans the space and is linearly independent. Different bases of the same finite-dimensional space contain the same number of vectors; this number is its dimension.
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