Consider the basis $\left{\vec{u}{i}\right}$ for $\mathbb{R}^{3}$, where $$ \vec{u}{1}=(1,0,0), \quad \vec{u}{2}=(1,1,0) \quad, \quad \vec{u}{3}=(1,1,1)
$$
are the components of these basis vectors when referred to the standard Cartesian basis $\left{\vec{e}{a}\right}$ whose components are tautologically $$ \vec{e}{1}=(1,0,0) \quad, \quad \vec{e}{2}=(0,1,0) \quad, \quad \vec{e}{3}=(0,0,1)
$$
Using the basis $\left{\vec{u}{i}\right}$, one can expand vectors as $\vec{v}=\alpha^{1} \vec{u}{1}+\alpha^{2} \vec{u}{2}+\alpha^{3} \vec{u}{3} .$ Consider a linear transformation $T$ that acts on $\vec{v}$ according to
$$
T: \vec{v} \rightarrow A \vec{v} \Longleftrightarrow \quad\left(\begin{array}{ll}
\left.\alpha^{1} \alpha^{2} \alpha^{3}\right)
\end{array}\left(\begin{array}{l}
\vec{u}{1} \ \vec{u}{2} \
\vec{u}{3} \end{array}\right)=\left(\begin{array}{lll} \alpha^{1} & \alpha^{2} & \alpha^{3} \end{array}\right)\left(\begin{array}{lll} 1 & 3 & 0 \ 0 & 2 & 3 \ a & 0 & 0 \end{array}\right)\left(\begin{array}{l} \vec{u}{1} \
\vec{u}{2} \ \vec{u}{3}
\end{array}\right)\right.
$$
where $a$ is a real number.
(i) What is the rank of $T$, defined as the dimension of the image space $T \mathbb{V} ?$ Does this depend on the choice of $a ?$ What is $\operatorname{det}(T) ?$ What is $\operatorname{tr}(T) ?$ When is this map an isomorphism? Justify your answers.
(ii) What is the matrix representation of $T$ in the Cartesian basis $\left{\vec{e}{a}\right} ?$ Call this matrix $A^{\prime} .$ Compute $\operatorname{det}\left(A^{\prime}\right)$ and $\operatorname{tr}\left(A^{\prime}\right) .$ The utility of the matrix and trace is that they are basisindependent quantities. (iii) These two basis choices are related by a linear transformation $S$ that maps one basis to the other: $\vec{e}{a}=S_{a}{ }^{i} \vec{u}_{i} .$ Find $S$ and show that it has an inverse. Construct the inverse transformation, and show that
$$
A^{\prime}=S A S^{-1}
$$
In words: The linear transformation in the new basis is the same as mapping from the new basis to the old one, applying $A$ there, and mapping back to the new basis.
(v) The lectures discussed the definition of a group, and a few examples. The set of invertible linear transformations of a vector space $\mathbb{V}$ parametrizes all possible choices of basis for $\mathbb{V}$. Any two bases are related by some invertible linear transformation. Show that these transformations form a group. This group is called $G L(\mathbb{V})$, the general linear transformations of $\mathbb{V} .$ Show that if $\operatorname{dim}(\mathbb{V})>1$, this group is non-abelian $-$ that is, there are group elements that have non-trivial commutator.

The Gram-Schmidt procedure and projection onto hyperplanes
Let $\left{\vec{u}{i}\right}, i=1, \ldots, n$ be a basis for a vector space $\mathbb{V}$ over $\mathbb{F}(\mathbb{R}$ or $\mathbb{C}) .$ The Gram-Schmidt procedure is a way of inductively constructing an orthonormal (ON) basis $\left{\vec{v}{a}\right}, a=1, \ldots, n$ from the $\left{\vec{u}_{i}\right}$ :

• Let $\vec{w}{1}=\vec{u}{1}$ and set $\vec{v}{1}=\vec{w}{1} /\left|\vec{w}_{1}\right|$
• Let $\vec{w}{j}=\vec{u}{j}-\sum_{a=1}^{j-1}\left\langle\vec{v}{a}, \vec{u}{j}\right\rangle \vec{v}{a}$, and then set $\vec{v}{j}=\vec{w}{j} /\left|\vec{w}{j}\right|$
  In this problem, we are going to show that the successive steps of the orthogonalization procedure involve projection onto hyperplanes spanned by subsets of the $\left{\vec{v}{i}\right}$. (i) A projection operator $P$ is a linear operator having the property $$ P \circ P=P $$ or in other words, if $M$ is the matrix representing $P$ in a basis, $M^{2}=M \cdot M=M .$ Consider the linear operators $P^{(a)}$ whose matrix elements in the $\left{\vec{u}{i}\right}$ basis are
  $$
  \left(M^{(a)}\right){i}^{j}=\sum{b=1}^{a} g_{i k}\left(\vec{v}{b}\right)^{k}\left(\vec{v}{b}\right)^{j}
  $$
  where $\left(\vec{v}{b}\right)^{j}$ is the $j^{\text {th }}$ component of the vector $\vec{v}{b}$ in the original basis $\left{\vec{u}{i}\right}$, or in other words ; $$ \vec{v}{b}=\left(\vec{v}{b}\right)^{j} \vec{u}{j}
  $$
  Also in the above the components of the metric tensor $\mathbf{g}$ (in the basis $\left.\left{\vec{u}{i}\right}\right)$ are defined as $$ g{i j}=\left\langle\vec{u}{i}, \vec{u}{j}\right\rangle
  $$
  (we can always define a tensor, which again like a vector is a notion defined independent of a choice of basis, by nevertheless choosing a basis and specifying its components in that basis). Show that $M^{(a)}$ is a projection operator, and that the rank of this matrix is $a$.
  (ii) Show that the operator $M^{(a)}$ acts as the identity operator on any vector lying in the span of $\left{\vec{v}{1}, \ldots, \vec{v}{a}\right} .$ Explain why $M^{(a)}$ is the operator that projects onto the hyperplane in $\mathbb{V}$ spanned by these vectors. Show that the image $\mathbb{W}$ of a projection operator is a subspace of $\mathbb{V}$ that is itself a vector space.
  (iii) Show that
  $$
  N^{(a)}=\mathbb{1}-M^{(a)}
  $$

It is sometimes useful to describe physics in a frame that is comoving with some particular object. For instance, we often forget that our “fixed” Kersten laboratory frame co-rotates with the Earth; sometimes it is important to correct for the fact that our frame of reference is not fixed, in fact it is not even moving at fixed velocity but is rather undergoing circular motion and thus accelerating. The importance of the distinction between a vector (position, velocity, acceleration, force) and its description in a particular basis as a triplet of numbers, are neatly illustrated in this problem.

Consider the two-dimensional version of this problem – e.g. a child on a carousel or merrygo-round. The child chooses to describe physics in the frame co-rotating with the platform of the merry-go-round. Let $\vec{v}{1}, \vec{v}{2}$ be the standard ON basis fixed with respect to the earth (whose rotation will be ignored for the purpose of this problem), with components $(1,0)$ and $(0,1)$ respectively. An (orthonormal) frame of reference rotating at angular velocity $\omega$ can be written as
$$
\vec{u}{1}=\cos \omega t \vec{v}{1}+\sin \omega t \vec{v}{2}, \quad \vec{u}{2}=\cos \omega t \vec{v}{2}-\sin \omega t \vec{v}{1}
$$
Note that if we write out the linear transformation between frames, it is a rotation by angle $\omega t$, that is the angle of rotation increases linearly in time (i.e. with constant angular velocity $\omega$ ).
(i) Let the position of an object be described in the fixed frame as
$$
\vec{x}(t)=\beta^{1}(t) \vec{v}{1}+\beta^{2}(t) \vec{v}{2}
$$
and in the rotating frame as
$$
\vec{x}(t)=\alpha^{1}(t) \vec{u}{1}(t)+\alpha^{2}(t) \vec{u}{2}(t)
$$
In other words, we are free to use any basis we want to describe a given vector, and we are also free to use different bases at different times in our description of the position vector, so long as we remember the time-dependence of the basis.

Consider an object undergoing inertial motion $\vec{x}(t)=\vec{x}{0}+\vec{v}{0} t$, where $\vec{x}{0}, \vec{v}{0}$ are fixed vectors (i.e. have constant, $t$-independent components in the fixed frame). What are the components of $x(t)$ in the fixed frame? In the rotating frame? Sketch the motion in the $\alpha^{1}-\alpha^{2}$ coordinate plane.
(ii) Consider Newton’s law of motion $\vec{F}=m \ddot{\vec{x}}$ in the rotating frame. Along the way you will encounter the time derivatives of the basis vectors $\vec{u}{i}$; however, since the $\vec{u}{i}$ are a basis, time derivatives such as $\partial_{t} \vec{u}{i}$ can be re-expressed as a linear combination of the $\vec{u}{i}$ themselves. Find the equations of motion for the coordinates $\alpha^{i}(t)$ of the object in the rotating frame, remembering to take into account that the frame of reference is itself time-dependent. Show that the effect of the rotating frame is to introduce a “fictitious force” (the centrifugal force) in the dynamics expressed in these coordinates.
(iii) Explain quantitatively why it is a good approximation to neglect the Earth’s circular motion and concentrate on that of the merry-go-round when analyzing the rotating frame effects.
(iv) An astronaut orbiting the Earth to a good approximation feels “weightless” as a result of the cancellation of the force of gravity and the “fictitious force” of the frame fixed with respect to their spaceship. Find the relationship between the Newtonian gravitational force on a spaceship in a circular orbit around the Earth at radius $r$, and the “fictitious force” that appears in the frame of the spaceship.

Einstein’s key insight in 1907 on the way to formulating his general theory of relativity was the equivalence principle: That the force of gravity can locally be traded off against being in an accelerated frame of reference. He then reasoned that the force of gravity itself is completely equivalent to using a local accelerated frame of reference, and can be expressed in terms of the way that such local frames compare from one point to another. Curved trajectories of test bodies in a gravitational field then led to curved geometry of space and time, and Newtonian gravity was replaced by the curved geometry of spacetime (after eight more years of hard work).