University Maths:Linear Alge.

還不明白的薑

2010-12-17 10:03 pm

Consider w1=(2,3,8,9) and w2=(9,0,-4,6) in R^4.

(i) Let x=(x1,x2.x3.x4).
Show that the condition that x is in span(w1,w2) is equivalent to a linearly independent system of 2 linear equations in x1,x2,x3,x4.

(ii) Use (i) to give a vector w3 in R^4 such that {w1,w2,w3} is linearly independent .

(iii) Use a determinant to double-check that your {w1,w2,w3} in (ii) is linearly independent.

更新1:

Let x=(x1,x2,x3,x4).

回答 (1)

myisland8132

2010-12-17 10:41 pm

✔ 最佳答案

(i) If x is in span(w1,w2), then αw1+βw2=x. Expand in details

x1=2α+9β,x2=3α,x3=8α-4β=4(2α-β),x4=3(3α+2β)

So, using x1 and x2 to represent α and β, we find that

12x1-80x2+27x3=0 and 6x1+23x2-27x4=0. We just show that the condition that x is in span(w1,w2) is equivalent to a linearly independent system of 2 linear equations in x1,x2,x3,x4.

(ii) Since a vector w in R^4 such that {w1,w2,w} is linearly independent, It cannot be represented by the form αw1+βw2. So if we set w1=w2=1 and then choose w3 and w4 such that 12w1-80w2+27w3 ≠0 and 6w1+23w2-27w4 ≠0, then w will satisfy the job. For example w3=w4=1. So w=(1,1,1,1).

(iii) If {w1,w2,w} in is linearly independent. Then the corresponding determinant formed by the sub-elements ≠0. For our case,
Choose 1st,2nd, 3rd elements. We find that

| 2 9 0 |
| 3 0 0 | =-27
| 8 -4 1 |

So, {w1,w2,w} in is linearly independent.

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