Q4. Suppose that $P$ is an $n \times n$ matrix such that $P^{2} = P$. Show that $\mathbb{R}^{n}$ is the direct sum of the range $R(P)$ and the nullspace $N(P)$ of $P$. Show also that $P$ represents the projection from $\mathbb{R}^{n}$ onto $R(P)$.
A4. Again – not sure how to start. Its idemptotent... so???
Q5. Prove that for any real matrix $A$, $N(A) = \text{orthogonal complement of } \ R(A^{T})$. Prove that for any real matrix, $N(A^{T}) = \text{orthogonal complement of} \ R(A)$.
A5. I get the first bit of this. It’s the second part I’m not sure about. Let $A^{\ast} = A^{T}$. Let $R(A)^{\ast}$ be the orthogonal complement of $R(A)$. What I said was let $x \in N(A)$. Then $Ax = 0$. So $A^{\ast}(Ax)=A^{\ast}0=0$. So $A^{\ast}(Ax)=0$. So $A(A^{\ast}x)=0$ so $A^{\ast}x=0$. So x is in the nullspace of $A^{\ast}$. I’m not sure about show to show the row space bit.
Q6. Let $A$ be a real matrix and let $R(A)$ denote its range. Show that the projection of a vector $b$ onto $R(A)$ parallel to the orthogonal complement of $R(A) – R(A)^{\ast}$ - is the vector in $R(A)$ closest to b.
Let A be a real matrix and let R(A) and R(A)* denote the range of A and the orthogonal complements of R(A) respectively. Show that the projection of a vector s onto R(A) parallel to R(A)* is the vector in R(A) closest to s.
A6. I think both of these questions are the same right? No idea where to start!!! :(
Im guessing these are all similar proofs - hence its on one thread.