thanks In Exercise 13a-c, how does the coefficient matrix of the system you solv

Question

thanks

In Exercise 13a-c, how does the coefficient matrix of the system you solved to find a basis for S relate to the vectors in S? Use the rank-nullity theorem to prove that if S = (x1,. . .Xk} is a linearly independent set of elements in Rn, then dim S = n-k. Let S = (x1, . . .,Xk} be a set of elements in Rn. Prove that X Rn is orthogonal to each element of S if and only if X is orthogonal to each element in the span of S-that is, if W = span S, then W = S Let W be a subspace of Rn. Prove that dim W + dim W = n. [Hint: Choose a basis (x1,. . .,Xk} for W and apply the results of Exercise 16 and 17.]

Explanation / Answer

let basis of W={X1,.....,Xk }

=>dimW=k.

From qs 17 its clear W(perpendicular)=S(perpendicular) =>dim(W(perpendicular)=dim(S(perpendicular)).

From qs 16 we get dim((S(perpendicular)))=n-k

Therefore dim(W)+dim(W(perpendicular)=k+n-k=n

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