2053Lecture2

# Vector spaces vector let 0 denote the zero vector a

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Unformatted text preview: ; are called basis vectors, and Ax, Ay and Az are the x, basis and y and z–components (scalars). The basis components vectors represent fundamental directions in 3– dimensional space, and any vector can be dimensional written in the above form. Vector Spaces Vector Let 0 denote the zero vector. A vector space Let vector is the set of vectors together with an addition and a scalar multiplication subject to the following axioms: following (A + B) + C = A+ (B + C) C) c (A + B ) = c A + c B ( c + d ) A = cA + d A (cd)A = c(dA) A+0=A (1)A = A A + (– A ) = 0 Scalar Multiplication Scalar Given a vector a, we can Given we produce a “new” vector by using scalar multiplication. If we multiply by –1 then the direction of the “new” vector is opposite to the direction of a. If we multiply by 2 then If the length of the “new” vector is twice as long as the vector a. Two Interpretations of Vector Addition Addition Parallelogram Rule: Triangle Rule: Triangle...
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