# Download e-book for kindle: A First Course in Linear Algebra by Daniel Zelinsky and Samuel S. Saslaw (Auth.) By Daniel Zelinsky and Samuel S. Saslaw (Auth.)

ISBN-10: 1483229564

ISBN-13: 9781483229560

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The set of all linear combinations of one nonzero posi­ tion vector V (that is, the set of all scalar multiples of v ) is the 30 1. Vectors set of all position vectors of points on a line through the origin. If u is another vector, the set of all u + as ¿ ranges over all scalars is the set of position vectors of points on a line through the head of u . 10(d). 22. Show that the set of all vectors of the form [2χ, x + y,x - y'] is the set of all linear combinations of [ 2 , 1, 1 ] and [0, 1, — 1 ] .

XO) Closure: a u is another object in the set. (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.

8 u · (αν + 6w) = a ( u · v) + 6 ( u · w ) . 1. [ 5 , 0 , 3 ] · [2, - 1 , - 4 ] = 5(2) + 0 ( - l ) + 3( - 4 ) = 10 + 0 - 12 = - 2 . Thus [5, 0, 3 ] and [2, - 1 , - 4 ] make an obtuse angle with each other. 2. 0 ( - 6 ) = 10. 3. [4, 1, 1 ] · [ 1 , - 2 , - 2 ] = 4 - 2 - 2 = 0. 2. 4. Find the angle between the vectors ν = [5, 1, 1 ] and w = [ - 1 , 2, 4 ] : II V II II w II COSÖ = v w = -5 + 2 + 4 = 1. i. 5 degrees. 5. The cosine of the angle a positive a;-axis is ( 1 / | | ν | | ) ( v · i ) because products, ν · i = || ν || || i || cos a.