Linear Systems - Mathematics 214 - Fall 2007

Turn in problems 3 & 4.

1. Give an example of a 3 x 3 matrix A such that rref(A) has exactly two nonzero rows (with the third row all zeros), but row(A) is not spanned by the first two rows of A.
2. Give an example of a 3 x 4 matrix A such that the leading columns of rref(A) do not span col(A).
3. Let A be any matrix. Prove that the columns of rref(A) are linearly dependent if and only if the columns of A are linearly dependent. You may not use the Fundamental Theorem of Invertible Matrices, nor the fact that row operations do not changed linear dependence relations between columns --- the point of the problem is to prove this very fact.. Hint: The columns of A are linearly dependent iff Ax=0 has a nontrivial solution.
4. Let W be a subspace of R^n such that dim(W) = k. Prove that if v₁, ..., v_k are linearly independent vectors in W, then they span W.
   Hint: Let v_k+1 be an arbitrary vector in W.  Suppose we write each of the vectors v₁, ..., v_k, v_k+1 as a linear combination of a basis {w₁, ..., w_k} for W, and then copy all the coeffecients of these linear combinations into a (k+1) by k matrix. What does Section 3.5 say about the linear dependence or independence of the rows of a matrix with more rows than columns? Conclude that v_k+1 must be in the span of v₁, ..., v_k.

Note: These problems are for review purposes. They aren't  directly related to Section 5.1 (though Problem 4 is related, indirectly: it's a generalization of Example 5.2, where it says "any three linearly independent vectors in R^3 span R^3". In Problem 4 instead of R^3 we have a subspace of arbitrary dimension).

Because we're nearing the end of semester, I will every now and then assign a few "review" problems in order to refresh your memory as well as solidify your understanding of some of the older topics.