By Daniel Zelinsky and Samuel S. Saslaw (Auth.)
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Extra resources for A First Course in Linear Algebra
7. 4(a). 17. 9 The components of every vector ν (which were defined geo metrically in Section 2, and which are the coefficients when ν is expressed as (equivalent to) a linear combination of i, j , and k ) are also given by the formulas ν · i, ν · j , ν · k. PROOF. 6. 2. Compute the following dot products: [ 2 , 1 , 2 ] · [ 3 , 4 , — 7 ] , [1, 0, 3 ] . [0, 1, 4 ] , [ 1 , V2, - 1 ] . [V2, 4, 0 ] , (2i - j + k ) · (i + 2j - k ) , ( 2 i - j ) . (j + k ) . 2. 1. 3. Which of the following vectors are perpendicular to which: 2i - j ; i; i + 2 j ; k; i + 2j + k?
Vectors V · w = II V II times the component of w along v . 4(b), because the vector sum (projection of w ) + (projection of w ' ) equals the projection of ( w + w ' ) , and so the scalar sum (component of w along v ) + (component of w ' along v ) equals the component of ( w + w ' ) along v . Multi plying this last equation by || ν ||, we get v « w + v * w ' = v* (w + w'). 4(a), thus: ( v + v ' ) · w = w · (v + v') = w · ν + w · v' = V · w + v' · w. (d) V · ( ( w + w ' ) + w " ) = V · ( w + w ' ) V · w + ν · w ' + V · w " , and similarly for ( e ) .
XI) a ( u - t - v ) =au (X2) (a + 6 ) u = a u + 6u. (X3) (afe)u = a ( 6 u ) . (X4) lu=u. + ay. 9. Still More General Vector Spaces 51 These axioms amomit to the statement that the ordinary arithmetic properties of addition and multiplication by scalars will work for the addition and multiplication of the objects in the vector space. In the preceding sections, we verified these axioms (usually explicitly, but sometimes implicitly) in the following two examples: (1) The vector space whose objects are arrows in 3-space with tail at the origin, with the addition and multiplication by scalars defined geometrically in Sections 4 and 5.