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taurus [48]
3 years ago
15

4. Let B = {(1,0,1),(0,1,2), (2, -1,0)}. (a) The vectors in B are dependent. Show this. (b) Find a non-trivial solution to an(1,

0,1) + a2(0,1,2) + a3(2, -1,0) = 0. (c) B is not a basis for the span(B). Find a basis for the span span(B).
Mathematics
1 answer:
Vladimir79 [104]3 years ago
3 0

Answer:

Step-by-step explanation:

Given are three vectors in set B.

To show that B is dependent

The determinant

\left[\begin{array}{ccc}1&0&1\\0&1&2\\2&-1&0\end{array}\right] \\=1(2)-1(2) =0

Hence vectors are dependent

b) The given equation a_1(1,0,1) + a_2(0,1,2) + a_3(2, -1,0) = 0.\\

Let us try parametrically

These 3 vectors are collinear and hence equation would be

\frac{x-1}{0-1} =\frac{y-0}{1} =\frac{z-1}{2-1} =s\\(x,y,z) = (1-s, s, 1+s)

c) Basis for B would be only 2 dimensional

i.e. any two vectors out of 3 form basis

The basis would be (1,0,1) and (0,1,2)

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