1 2 Determine if each set is linearly independent in the nat

1.

2. Determine if each set is linearly independent (in the natural vector space).

PROBLEM 3.3. Is the vector 0in the span of the set 1 -1)?

Solution

Only one problemcan be answered at a time. Hence problem 1 is being answered as follows:

(1,0,3) will lie in the span of (2,1,-1) and (1,-1,1) if it is some linear combination of these vectors. Assume it lies in the span, then C₁ (2,1,-1) +C₂ (1,-1.1)= (1,0,3) . Comparing both sides,

2C₁ +C₂ =1; C₁ -C₂=0 and -C₁ +C₂=3

It is seen from the above that C₁=C₂ and C₂= 3+C_1. Now, C₂ can not be equal to C₁ and also 3+C₁ at the same time, unless both C₁ and C₂ are identically 0. This proves that (1,0,3) is linearly independent and does not lie the span of (2,1,-1) and (1,-1,1)

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