91Ó°ÊÓ

What matrix has the effect of rotating every vector through \(90^{\circ}\) and then projecting the result onto the \(x\)-axis? What matrix represents projection onto the \(x\)-axis followed by projection onto the \(y\)-axis?

What is the smallest subspace of 3 by 3 matrices that contains all symmetric matrices and all lower triangular matrices? What is the largest subspace that is contained in both of those subspaces?

Every straight line remains straight after a linear transformation. If \(z\) is halfway between \(x\) and \(y\), show that \(A z\) is halfway between \(A x\) and \(A y\).

Show that the set of nonsingular 2 by 2 matrices is not a vector space. Show also that the set of singular 2 by 2 matrices is not a vector space.

(Recommended) If we add an extra column \(b\) to a matrix \(A\), then the column space gets larger unless _________. Give an example in which the column space gets larger and an example in which it doesn't. Why is \(A x=b\) solvable exactly when the column space doesn't get larger by including \(b\) ?

If the entries of a 3 by 3 matrix are chosen randomly between 0 and 1 , what are the most likely dimensions of the four subspaces? What if the matrix is 3 by \(5 ?\)

Suppose \(A\) is an \(m\) by \(n\) matrix of rank \(r\). Its reduced echelon form is \(R\). Describe exactly the reduced row echelon form of \(R^{\mathrm{T}}\) (not \(A^{\mathrm{T}}\) ).

Construct a 2 by 2 matrix whose nullspace equals its column space.