thanks Suppose that {X1, X2} is a linearly independent set of elements in some v
ID: 2968363 • Letter: T
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Suppose that {X1, X2} is a linearly independent set of elements in some vector space V. Let Y1 = aX1 + bX2 and Y2 = cX1 + dX2. Prove that Prove that {Y1, Y2} is linearly independent in V if and only if the vectors [a, b]t and [c, d]t are linearly independent elements of R2. Create an exercise similar to Exercise 12, using different coefficients for the Xi, in which Y1 and Y2 turn out to be linearly dependent. Create an exercise similar to Exercise 12, using different coefficients for the Xi, in which Y1 and Y2 turn out to be linearly independent. Suppose that {X1,X2} is a linearly independent set of elements in some vector space V. Let Y1 = 3X1 - 2X2 and Y2 = X1 + X2.Prove that {Y1,Y2} is linearly independent in V.Explanation / Answer
13.
(a)
Y1 and Y2 are linearly independent
iff s.Y1+t.Y2 = 0 => s = 0 = t
iff s.(aX1+bX2)+t.(cX1+dX2) = 0 => s = 0 = t
iff (sa+tc).X1+(sb+td).X2 = 0 => s=0=t
iff sa+tc = 0 = sb+td => s = 0 = t (since p.X1 + q.X2 = 0 iff p = 0 = q)
iff ad-bc is nonzero
iff (a;c) and (b;d) are linearly independent vectors.
(b)
Y1 = X1+2X2
Y2 = 2X1+4X2.
Then Y1 and Y2 are linearly dependent since (1; 2) and (2; 4) are linearly dependent.
(c)
Y1 = X1+2X2
Y2 = 2X1+3X2.
Then Y1 and Y2 are linearly independent since (1;2) and (2;3) are linearly independent.
12.
Let a.Y1+b.Y2 = 0
Then a.(3X1-2X2)+b.(X1+X2) = 0
=> (3a+b).X1+(-2a+b).X2 = 0
=> 3a+b = 0 = -2a+b (since X1,X2 are linearly independent)
=> b = 2a, 5a = 0 (substituting b obtained from second eqn in first equation)
=> a = 0 = b.
Thus a.Y1+b.Y2 = 0
=> a = 0 = b.
Thus Y1 and Y2 are linearly independent.
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