Linear Algebra vector problems?

Given the vectors u =(1,2,3) and v = (2,3,1) in R3:

a. Find k so that w = (1,k,4) is a linear combination of u and v. [5]

b. Discuss the span of u, v, and w. [5]

Comments

  • au = (a, 2a, 3a)

    bv = (2b 3b, b)

    a+2b = 1

    2a + 3b = k

    3a +b = 4

    b = 4-3a

    a + 2(4-3a) = 1

    a+ 8 -6a = 1

    -5a = -7

    a = 7/5

    b = 4 - 3(7/5) = 4-21/5 = -1/5

    k = 2a + 3b = 14/5 -3/5 = -11/5

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