Find a matching answer for each of the following questions. Use the APP \"5B Bas

Question

Find a matching answer for each of the following questions. Use the APP "5B Basis and Coordinates in R "on C4 to check your answers. Choosev Given the basis B V1.V2? for R where v(-4,2) and vz (1,2), find a match for the coordinate vector of X (16,2) relative to B Given the basis B-(V.V2) for R2 where V (4.2) and v (1.2) ind a match for X given that the coordinate vector of X relative to B is (3-2) Choose.. Given the basis B-Vi.V2 for R where Vi-(-4,2) and v (1,2), find a match for the coordinate vector of x- (-14,-8) relative to B Choose...v

Explanation / Answer

Given the basis B = {(-4,2),(1,2)}.

a) Let us consider a relation X = a(-4,2)+b(1,2) where a,b are real numbers.

Then, (16,2) = a(-4,2)+b(1,2)

i.e., (16,2) = (-4a+b,2a+2b)

i.e., -4a+b = 16

2a+2b = 2

Solving we get, a = -3 and b = 4.

Therefore, coordinate vector of X relative to B is (a,b) = (-3,4).

b) Given, coordinate vector of X relative to B is (-3,-2).

Then, X = (-3)(-4,2)+(-2)(1,2) = (12-2,-6-4) = (10,-10)

Therefre, match for X is (10,-10).

c)

Let us consider a relation X = a(-4,2)+b(1,2) where a,b are real numbers.

Then, (-14,-8) = a(-4,2)+b(1,2)

i.e., (-14,-8) = (-4a+b,2a+2b)

i.e., -4a+b = -14

2a+2b = -8

Solving we get, a = 2 and b = -6.

Therefore, coordinate vector of X relative to B is (a,b) = (2,-6).

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