Projection Into a Subspace

Find $M^{\perp}$.

• (a)  $M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\end{pmatrix}\mid x+y=0\hskip 1.00006pt\}$}$     (b)  $M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\end{pmatrix}\mid-2x+3y=0\hskip 1.00006pt\}$}$     (c)  $M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\end{pmatrix}\mid x-y=0\hskip 1.00006pt\}$}$     (d)  $M=\text{$\{\hskip 1.00006pt\vec{0}\,\hskip 1.00006pt\}$}$     (e)  $M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\end{pmatrix}\mid x=0\hskip 1.00006pt\}$}$     (f)  $M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mid-x+3y+z=0\hskip 1.00006pt\}$}$     (g)  $M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mid\text{$x=0$ and $y+z=0$}\hskip 1.00006pt\}$}$

Find the orthogonal projection of the vector into the subspace.

With the same subspace as in the prior problem, find the orthogonal projection of this vector.

Let $\vec{p}$ be the orthogonal projection of $\vec{v}\in\mathbb{R}^{n}$ onto a subspace $S$. Show that $\vec{p}$ is the point in the subspace closest to $\vec{v}$.

This subsection shows how to project orthogonally in two ways, the method of Example 3.2 and 3.3, and the method of Theorem 3.8. To compare them, consider the plane $P$ specified by $3x+2y-z=0$ in $\mathbb{R}^{3}$.

• Find a basis for $P$.

• Find $P^{\perp}$ and a basis for $P^{\perp}$.

• Represent this vector with respect to the concatenation of the two bases from the prior item.

• Find the orthogonal projection of $\vec{v}$ into $P$ by keeping only the

We have three ways to find the orthogonal projection of a vector into a line, the Definition 1.1 way from the first subsection of this section, the Example 3.2 and 3.3 way of representing the vector with respect to a basis for the space and then keeping the $M$ part, and the way of Theorem 3.8. For these cases, do all three ways.

• $\vec{v}=\begin{pmatrix}1\\ -3\end{pmatrix},\quad M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\end{pmatrix}\mid x+y=0\hskip 1.00006pt\}$}$

• $\vec{v}=\begin{pmatrix}0\\ 1\\ 2\end{pmatrix},\quad M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mid\text{$x+z=0$ and $y=0$}\hskip 1.00006pt\}$}$

Check that the operation of Definition 3.1 is well-defined. That is, in Example 3.2 and 3.3, doesn’t the answer depend on the choice of bases?

What is the orthogonal projection into the trivial subspace?

What is the projection of $\vec{v}$ into $M$ along $N$ if $\vec{v}\in M$?

Show that if $M\subseteq\mathbb{R}^{n}$ is a subspace with orthonormal basis $\langle\vec{\kappa}_{1},\ldots,\vec{\kappa}_{n}\rangle$ then the orthogonal projection of $\vec{v}$ into $M$ is this.

Prove that the map $p\colon V\to V$ is the projection into $M$ along $N$ if and only if the map $\operatorname{id}-p$ is the projection into $N$ along $M$. (Recall the definition of the difference of two maps: $(\operatorname{id}-p)\,(\vec{v})=\operatorname{id}(\vec{v})-p(\vec{v})=\vec{v}-p(\vec{v})$.)

Show that if a vector is perpendicular to every vector in a set then it is perpendicular to every vector in the span of that set.

True or false: the intersection of a subspace and its orthogonal complement is trivial.

Show that the dimensions of orthogonal complements add to the dimension of the entire space.

Suppose that $\vec{v}_{1},\vec{v}_{2}\in\mathbb{R}^{n}$ are such that for all complements $M,N\subseteq\mathbb{R}^{n}$, the projections of $\vec{v}_{1}$ and $\vec{v}_{2}$ into $M$ along $N$ are equal. Must $\vec{v}_{1}$ equal $\vec{v}_{2}$? (If so, what if we relax the condition to: all orthogonal projections of the two are equal?)

Let $M,N$ be subspaces of $\mathbb{R}^{n}$. The perp operator acts on subspaces; we can ask how it interacts with other such operations.

• Show that two perps cancel: $(M^{\perp})^{\perp}=M$.

• Prove that $M\subseteq N$ implies that $N^{\perp}\subseteq M^{\perp}$.

• Show that $(M+N)^{\perp}=M^{\perp}\cap N^{\perp}$.

The material in this subsection allows us to express a geometric relationship that we have not yet seen between the range space and the null space of a linear map.

• Represent $f\colon\mathbb{R}^{3}\to\mathbb{R}$ given by

  with respect to the standard bases and show that

  is a member of the perp of the null space. Prove that $\mathscr{N}(f)^{\perp}$ is equal to the span of this vector.

• Generalize that to apply to any $f\colon\mathbb{R}^{n}\to\mathbb{R}$.

• Represent $f\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$

  with respect to t…

Define a projection to be a linear transformation $t\colon V\to V$ with the property that repeating the projection does nothing more than does the projection alone: $(t\mathbin{\circ}t)\,(\vec{v})=t(\vec{v})$ for all $\vec{v}\in V$.

• Show that orthogonal projection into a line has that property.

• Show that projection along a subspace has that property.

• Show that for any such $t$ there is a basis $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$ for $V$ such that

  where $r$ is the rank of $t$.

• Conclude that every pr…

A square matrix is symmetric if each $i,j$ entry equals the $j,i$ entry (i.e., if the matrix equals its transpose). Show that the projection matrix $A({{A}^{\mathsf{T}}}A)^{-1}{{A}^{\mathsf{T}}}$ is symmetric. Hint. Find properties of transposes by looking in the index under ‘transpose’.