Problems in class

1. Solutions of linear systems Let \(A\) be a \(m\times n\) matrix, and let \({\bf b}\) be a \(m\times 1\) vector. Show that the set of all solutions of \(A{\bf x} = {\bf b}\) (where \(\bf x\) is considered unknown) is

   \[S= \{ {\bf x}_p +{\bf z} : {\bf z} \in \textrm{ null}(A) \},\]

   where \({\bf x}_p\) is (any) one solution of \(A{\bf x}={\bf b}\) (often called a particular solution). In particular, note that the set \(S\) remains the same no matter which particular solution we choose.

   This reflects what you have seen in EE213 and will see in other classes for solving differential equations. The analogy (sum of particular and homogenous parts) is no accident. We can construct linear spaces out of functions, just as we are doing with vectors in EE 345. Depending on how we construct them, they have inner products, norms, etc. just like vectors, and certain spaces of functions reflect the geometry of the spaces we are studying almost perfectly (Hilbert spaces). But the analogy requires care—many times, spaces of functions we construct have infinite dimension (like a Fourier basis), and this requires some technical care.

2. Suppose \(A\) is a \(m\times n\) matrix.

   • If \(C\) is a \(n\times n\) matrix, show that the column space of \(AC\) is a subset of the column space of \(A\)
   • Show that if \(C\) is a \(n\times n\) invertible matrix, then the column space of \(AC\) equal to the column space of \(A\)? Give and example to show that the condition on \(C\) is not always necessary
   • What is the column space of \(\begin{bmatrix} A & AC \end{bmatrix}\)?
   • Can you make statements analogous to the above about row spaces?
   • The (row/column) space is preserved in each step of the Gaussian elimination procedure. (choose the correct answer, with an explanation)

3. Linear Regression This is an important result which we will need for linear regression in a future module. Show that the null space of a \(n\times p\) matrix \(X\) equals the null space of \(X^TX\).

4. Let \(A\) have rref \(R_1\), and let \(A^T\) have rref \(R_2\). We will relate the fundamental spaces of \(A\) to those of \(R_1\) and \(R_2\).

   • Convince yourself that \(R_1\) and \(R_2\) are not transposes of each other in general.
   • The column space of \(A\) equals the (row/column) space of (\(R_1/R_2\))? (choose correct answers with explanations)
   • The row space of \(A\) equals the (row/column) space of (\(R_1/R_2\))? (choose correct answers with explanations)
   • The right null space of \(A\) equals the (left/right) null space of (\(R_1/R_2\))? (choose correct answers with explanations)
   • The left null space of \(A\) equals the (left/right) null space of (\(R_1/R_2\))? (choose correct answers with explanations)

5. These problems on stacked matrices recap the basic concepts behind Gaussian elimination and the linear spaces we have learnt about.

   • If \(A\) is an invertible 4x4 matrix, what is the rref of \(\begin{bmatrix} A & A \end{bmatrix}\) and that of \(\begin{bmatrix} A\\ A \end{bmatrix}\)? What about \(\begin{bmatrix} A & AB \end{bmatrix}\)? (You do not need to know \(A\) to write the answers out, express the last part in terms of \(B\)).

   • Express the (right) null space of \(\begin{bmatrix} A\\ B \end{bmatrix}\) in terms of the right null spaces of \(A\) and \(B\).