Inner products, orthogonality, and orthonormal bases from a college linear algebra course.
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- Definition of inner product
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A function on two vectors returning a scalar with properties: linearity, symmetry, and positive definiteness- Standard inner product in R^n
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u dot v = u_1*v_1 + u_2*v_2 + ... + u_n*v_n- What does bilinearity of an inner product mean?
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It is linear in each argument: <au+bv, w> = a<u,w> + b<v,w> and similarly in the second slot- Symmetry property of inner product
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<u,v> = <v,u>- Positive definiteness of inner product
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<v,v> > 0 when v is nonzero, and <v,v> = 0 if and only if v = 0- When are two vectors orthogonal?
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When their inner product is zero: <u,v> = 0- Is the zero vector orthogonal to other vectors?
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Yes, the zero vector is orthogonal to every vector- What is an orthonormal set?
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A set where vectors are pairwise orthogonal and each has unit norm (length 1)- What is an orthonormal basis?
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A basis consisting of vectors that are pairwise orthogonal and each has unit norm- How do you compute the norm of a vector v?
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||v|| = sqrt(<v,v>)- Distance between vectors u and v
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d(u,v) = ||u - v|| = sqrt(<u-v, u-v>)- Formula for the angle theta between nonzero vectors u and v
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cos(theta) = <u,v> / (||u|| * ||v||)- First step of the Gram-Schmidt process
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Take the first vector v_1 and normalize it: e_1 = v_1 / ||v_1||- General step of the Gram-Schmidt process
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For vector v_i, subtract its projections onto all previous orthonormal vectors, then normalize the result- Orthogonal projection of u onto vector v
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proj_v(u) = (<u,v> / <v,v>) * v
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