Pset 6: Due Monday April 6
(1.) GS p. 161 Problem 24) Give examples of matrices A for which the number of solutions to Ax=b is
(a) 0 or 1, depending on b
(b) $\infty$, regardless of b
(c) 0 or $\infty$, depending on b
(d) 1, regardless of b
(2) Inspired by GS p163 Problem 34. Suppose A is 3x4 and the nullspace consists of multiples of s = (2,3,1,0).
(a) What are the dimensions of the four fundamental subspaces?
(b) How do you know that $Ax=b$ can be solved for all b?
(3) GS p.177 Problem 18. Suppose $v_1,v_2,\ldots,v_6$ are six vectors in $R^4$.
(a) Those vectors (do)(do not)(might not) span $R^4$.
(b) Those vectors (are)(are not) (might be) linearly independent.
(c) Any four of those vectores (are)(are not)(might be) a basis for $R^4$.
(4)Inspired by GS p.192 Problem 21. Under what possible conditions is the matrix $A=uv^T+wz^T$ not of rank 2?
(5) GS p.203 Inspired by Problem 10. If $A$ is symmetric, why is the column space perpendicular to the nullspace?
(6) GS p.202 Problem 4. If $AB=0$ then the columns of B are in the [--2 words--] of A. The rows of A are in the [--2 words--] of B. With AB=0, why can't A and B be 3x3 matrices of rank 2?
(7) GS p.204 Problem 24. Suppose an mxn matrix is invertible: $AA^{-1}=I$. Then the first column of $A^{-1}$ is orthogonal to the space spanned by which rows of $A$?
(8) A matrix is m x n what are the possible dimensions?
(A) dim(col(A))?
(B) dim(row(A))+dim(null(A))?
(C) the sum of the dimensions of the four fundamental subspaces?
(D) dim(col(A)) + dim(row(A))?
(9)Suppose y₁(x),y₂(x),y₃(x),y₄(x) are four non-zero polynomials of degree at most 2. (This means the functions have the form ax²+bx+c, where at least one of the coefficients is nonzero.) What possibilities are there in the dimension of the vector space spanned by y₁(x),y₂(x),y₃(x),y₄(x)? Give examples for each possibility and explain briefly why no other dimension can happen.
(10) A reflector is defined as a matrix of the form $Q=I−2uu^T$ where ‖u‖=1.
(A) Show that a reflector is orthogonal by showing that Q is symmetric and $Q^2=I$.
(B) Explain briefly why this makes Q orthogonal.