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Suppose vector a=[1,1,2], vector b=[2,0,3], vector c=[4,2,7], vector p=[0,1,1], vector q=[1,0,1], vector r=[1,1,0].

Determine whether vector v=[7,-2,9] belongs to :

(a) span {vector a, vector b, vector c}

(b) span {vector p, vector q, vector r}

I don't know how to find the span

FInding the span: To span 3-D, we need

1. vectors with 3 components (the x, y, z dimensions)

2. none of the components is zero in ALL of the vectors (otherwise they couldn't span that dimension)

3. at least 3 vectors that are LINEARLY INDEPENDENT.

Items 1 and 2 are obvious. We also meet the first requirement of Item 3 (3 vectors). The rest of Item 3 requires some calculation. To be linearly independent means (the short answer is) that the so-called determinant of the matrix consisting of the columns of the 3 vectors is zero. I assume you have been introduced to determinants, but in any case, the calculation goes like

det(a^t,b^t,c^t) where the superscript "t" means transpose (i.e., turns row vector into column vector)

= | 1 2 4 |

| 1 0 2 |

| 2 3 7 |

= 1･(0･7-2･3)-2･(1･7-2･2)+4･(1･3-0･2) <- not unlike finding a cross product

= -6 - 6 + 12 = 0.

So the vectors are linearly independent and so span 3-D. Since the vector v is 3-D, it "belongs" to the space spanned by a,b ,c.

Use the same procedure to analyze vectors p, q and r. Good luck.

Send me a follow-up if you want more details on the calculation (including theory).

Randy

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