Show that the set S = {(0, b, c) | b,.c e k} is a of RJ. Show that the set S = I

Question

Show that the set S = {(0, b, c) | b,.c e k} is a of RJ. Show that the set S = Is S {e1 (0, 1, 2, 5), (1. -1, 0, 3)} a basis for R4? Justify your answer. Is S = {(0, -2), (3, 4)} a basin for RJ? Justify your answer. Express the vector (7, 2) in R2 as a linear combination of the vectors (2, 5) and Express the vector t3 - 6 in Py as a linear combination of the vectors 2t3 + 3t2 - Show that the set B - {-t + 5, t2 - 8, t2 + 2t - 3} is a basis for Find a basis for R31 that includes the vectors v1 = (6, 2, -3) and v2 = (4, 1, 0). Find a basis for and the dimension of the solution space of the homogeneous system Find rank A and nullity A of the matrix A =

Explanation / Answer

Q4)
B = {-t+ 5, t2 - 8 , t2 + 2t -3}

P2 = a1x0 + a2x1 + a3x2

If we show that the three vectors os set B are independent, then they will imply that they span P2 and that B will be a basis for P2

SO,

c1( -t+5) + c2(t2-8) + c3(t2 + 2t - 3) = 0.t0 + 0.t1 + 0.t2

(5c1 - 8c2 - 3c3)t0 + (-c1 + 2c3)t1 + (c2 + c3).t2 = 0t0 + 0t1 + 0t2

5c1 - 8c2 - 3c3 = 0

-c1 + 2c3 = 0

c2 + c3 =0

When you solve you get c1 = c2 = c3 = 0

Hence the three vectors are independent.

This means that they span P2 also.

This means that the set B containing these 3 vectors is basis of P2

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