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Elina [12.6K]
4 years ago
11

Is it possible that vectors v1, v2, v3 are linearly dependent, but the vectors w1 = v1 + v2, w2 = v2 + v3 and w3 = v3 + v1 are l

inearly independent?
Mathematics
1 answer:
Jobisdone [24]4 years ago
6 0

Since v_1,\ v_2,\ v_3 are linearly dependent, there exist coefficients \alpha_1,\ \alpha_2,\ \alpha_3 such that

\alpha_1v_1+\alpha_2v_2+\alpha_3v_3=0

Now, a linear combination of the new vectors would look like this:

\beta_1w_1+\beta_2w_2+\beta_3w_3 = \beta_1(v_1+v_2)+\beta_2(v_2+v_3)+\beta_3(v_1+v_3)

Which simplifies to

(\beta_1+\beta_3)v_1+(\beta_1+\beta_2)v_2+(\beta_2+\beta_3)v_3

So, any linear combination of w_1,\ w_2,\ w_3 is also a linear combination of v_1,\ v_2,\ v_3. This implies that we can choose the coefficients for a linear combination that will give the zero vector.

In particular, if \alpha_1,\ \alpha_2,\ \alpha_3 are the coefficients such that

\alpha_1v_1+\alpha_2v_2+\alpha_3v_3=0

we can choose

\beta_1+\beta_3=\alpha_1,\quad \beta_1+\beta_2=\alpha_2,\quad \beta_2+\beta_3=\alpha_3

And we have

\beta_1w_1+\beta_2w_2+\beta_3w_3 = 0

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