The geometry behind adjoint of a linear transformation I was going through friedberg section 6.3 where they have introduced the term adjoint of a linear transformation. My basic question is what does it mean to have an adjoint of a linear transformation geometrically (just like we can imagine transformation as an operation on a vector which can rotate the vector or increase or decrease it's length) and why is it important to know?
 A: It is helpful to first consider the adjoint of a linear transformation $T: V \to V$ for an arbitrary vector space $V$.  Since we cannot use an inner product, the adjoint $T’: V^* \to V^*$ is defined on the dual space instead, and takes a linear functional $f: V \to \mathbb{R}$ to the linear functional $T’ f = f \circ T$ (in words, $f$ precomposed by $T$). Linear functionals can be thought of as coordinate functions, for example on $V = \mathbb{R^2}$ we have the functionals $x, y : \mathbb{R}^2 \to \mathbb{R}$ which read off the first and second entries in a vector respectively. I imagine a functional as a series of parallel hyperplanes, the solutions to $f(v) = \ldots -1, 0,1,2,\ldots$. Scaling $f$ to be larger will make the hyperplanes closer together.
We have $x(Tv) = (T’x)(v)$ for all vectors $v\in \mathbb{R}^2$ (by definition of the adjoint). This means that performing the transformation $T$ and then reading the $x$-coordinate is the same as just using the new functional $T’x$, which reads vectors “as if” they had already had $T$ applied to them. If $T$ is a scaling by 2 for instance, we are saying that $x(2v) = (2x)(v)$ - quite unsurprising, but treating $2x$ as a new object in its own right (the $x$ axis with double the number of ticks) can be quite useful, especially for more interesting operators $T$. For example, imagine you have a lot of vectors and you want to measure their $x$-coordinates before and after a rotation by 30°. One way would be to compute $x(v)$ and $x(Tv)$ for every vector, but another would be to compute $z = T’x$ once, and then compute $x(v)$ and $z(v)$ for every vector (much more efficient, requiring only dot products rather than matrix multiplications each time).
Finally, adjoints show up in inner product spaces, because there is a canonical bijection between vectors and linear functionals. To each vector $v$ we get a special linear functional $f_v(u) = (v, u)$, and for any linear transformation $T$ we can take the adjoint of $f_v$ as above, to get $(T’ f_v)(u) = f_v(Tu) = (v, Tu)$. Define a bijection $T^*: V\to V$ by the condition that $T’f_v = f_{T^* v}$, then it turns out that $T^*$ is linear, and moreover is the unique linear map such that $(T^* v, u) = (v, Tu)$ for all $u,v \in V$.
The adjoint in inner product spaces holds the same meaning as before: if $(u, -)$ is a “coordinate function”, then applying $T$ and measuring to get $(u, Tv)$ is the same as first applying $T^*$ to the coordinate function and then measuring to get $(T^* u, v)$. The significance is that since $T^*$ is now an operator on the original space (rather than the dual space), we can ask lots of interesting questions about how $T$ relates to its adjoint $T^*$.
