thanks Does the subspace W in Example 7 on page 119 contain (a) [9, -8, -5]t? (b
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Does the subspace W in Example 7 on page 119 contain (a) [9, -8, -5]t? (b) [1,2,3]t? Suppose that W is a two-dimensional subspace of R3 that contains the vectors X1 = [1,2, -3]t and X2 = [3, -7,2]t Does W contain y = [1,1,1]t? Since W is two-dimensional, any set of two linearly independent vectors in W spans W. Thus, {Xi, X2} spans W. The vector Y belongs to W if and only if there are scalars x and y such that xX1 + yX2 = Y. Substituting for Y, X1 and X2 we see that the vector equation is equivalent to the system x + 3 y =1 2x-7y =1 -3x + 2 y =1 We row reduce the augmented matrix, obtaining Thus, the system is inconsistent, showing that Y is not an element of W We close this is section with a rather sobering comment. All the vector spaces considered in .his section were finite-dimensional in the sense that they could be spanned by a finite number of elements. There do however, exist vector spaces so big that they cannot be spanned by any finite number of elements. Such spaces are called " infinite-dimensional". The simplest example is the space R infinity. This is by definition the set of all "vectors" of the form [x1,x2,xn, where {xn}infinity n=1, is an infinite sequence of real numbers Thus, for example, both X = [1,2,3,4,,n, and Y = [1,1/2,1/3,1/4,,1/n, represent elements of R infinity We can add elements of ; and multiply them by scalars just as we do for elements of Rn. Thus, for X and Y) as above X + Y= [2,2+ ½,3+ 1/3,4+ 1/4,.,n + 1/n,Similarly, 2X = [2,4,6,8,10,... ,2n,... It is easily seen that R infinity satisfies all the vector space properties listed on page 10. Thus, R infinity is a vector space. Let Ij be the element of lx that has a 1 in the jth position and O's in all other positions. Thus, for example, I3 = [0,0,1,0,0,0,... It is clear that each of the Ij is linearly independent of the other Ik, since each Ij, has a 1 in a position where all the others have a 0. Thus, R infinity has an infinite set of linearly independent vectors. This proves that R infinity is infinite-dimensional because (by Theorem 1) in an n-dimensional space we can have at most n linearly independent elements.Explanation / Answer
Actually, the quickest way to solve this is to determine if the determinant is 0. If so, as the first two vectors are independent, the third is not.
1 3 9
2 -7 -8
-3 2 -5
has determinant 0
Thus, W contains [9, -8, -5]t
In fact, we can easily see the 3(1, 2, -3)t + 2(3, -7, 2)t = (9, -8, 5)t
1 3 1
2 -7 2
-3 2 3
has determinant -78
Thus, the three vectors are linearly independent, so W does not contain (1, 2, 3)t
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