Inner product, length, and orthogonality - Orthogonality and Least Squares

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Inner product, length, and orthogonality

Lessons

Notes:
Let vectors uu and vv be:
vector u and vector v

Then the inner product of the two vectors will be:
inner product of vector u and vector v

Let u,vu,v and ww be vectors in Rn\Bbb{R}^n, and let cc be a scalar. Then,
1) uv=vuu \cdot v=v \cdot u
2) w(u+v)=wu+wv w(u+v)=w \cdot u+w \cdot v
3) (cu)v=c(uv)=u(cv) (cu) \cdot v=c(u \cdot v)=u \cdot (cv)
4) uu0u \cdot u \geq 0, and uu=0u \cdot u=0 only if u=0u=0

Suppose vector v. Then the length (or norm) of a vector vv is

v=v12+v22++vn2,v2=vv \lVert v \rVert = \sqrt{v_{1}^2+v_{2}^2+\cdots +v_{n}^2}\;,\; \lVert v \rVert ^2 = v \cdot v

Suppose dist(u,v)(u,v) is the distance between the vectors uu and vv. To find the distance between the two vectors, we calculate
dist(u,v)=uv(u,v)=\lVert u-v \rVert

If two vectors uu and vv are orthogonal to each other, then it must be true that
uv=0u \cdot v =0
  • Intro Lesson
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Inner product, length, and orthogonality

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