Linear Algebra Step by Step Notes Chapter 2

Posted on August 11, 2015 by Sumant Sumant

Proposition 2.1

Let u,v and w be vectors in R^n and k,c be real numbers (or real scalars). We have the following results

u+v = v+u

Proposition 2.2

Let u be a vector in R^n. Then the vectors -u which satisfies property $u+(-u)=0$ is unique

Proposition 2.3 Let u be a vector in R^n there is only one vector -u in R^n such that u+(-u)=0

Proposition 2.4 This matrix multiplication is called the dot product or inner product of the vectors u and v and

u.v= u^Tv=u_1v_1+u_2v_2+u_3v_3+...+u_nv_n

Proposition 2.5 Vectors u and v are perpendicular or orthogonal if and only if their dot product is zero. Two vectors u and v in R^n are said to be perpendicular or orthogonal \iff u \cdot v = 0

Proposition 2.6 Let u,v and w be vectors in R^n and k be a real scalar (real number). We have the following

Proposition 2.7 Let u be a vector R^n then the length of u = \begin{\bmatrix} u_1 \\ u_2 \end{\bmatrix} is given by

||u|| = \sqrt{u_1^2+u_2^2+...+u_n^2}

Proposition 2.14 Cauchy-Schwarz inequality

Let u and v be vectors in R^n then

|u \cdot v|=||u|| ||v||

Proposition 2.15 Minkowski (triangular inequality)

Let u and v be the vectors in R^n then

||u+v|| \le ||u||+||v||

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