Let V denote the space of polynomials in x of degree ≤ 3 over a field K. Let v1 =1+x+x2 +x3, v2 =1+2x+3x2, v3 =2+6x, v4 =6. Are the vectors v1, v2, v3, v4 linearly independent over each of K and explain why. (i) K=R; (ii) K = F3, the field of integers mod 3; (iii) K = F5, the field of integers mod 5.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let V denote the space of polynomials in x of degree ≤ 3 over a field K. Let
v1 =1+x+x2 +x3, v2 =1+2x+3x2, v3 =2+6x, v4 =6.

Are the vectors v1, v2, v3, v4 linearly independent over each of K and explain why.

(i) K=R;

(ii) K = F3, the field of integers mod 3;

(iii) K = F5, the field of integers mod 5.

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