Suppose your data matrix is $\mathbf{X} \in \mathbb{R}^{n \times m}$, where $m$ is the number of data points, and $n$ is the dimensionality of the data. Let the SVD of $\mathbf{X}$ be
\begin{align}
\mathbf{X} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}.
\end{align}
Assume for the sake of simplicity that the data points that make up $\mathbf{X}$ have been not been translated (for example, if you normally subtract the mean, assume it's zero). Also, assume that the singular values are in decreasing order from left to right. Let $\mathbf{U}_{k}$ be the matrix consisting of the leftmost $k$ columns of $\mathbf{U}$ (i.e., the left singular vectors corresponding to the $k$ largest singular values). Then PCA defines the following transformations:
- Mapping into a lower dimensional coordinate system: given $\mathbf{x} \in \mathbb{R}^{n}$, where $n$ corresponds to the dimensionality of your original data, the mapped (or transformed) data will be $\mathbf{y} \in \mathbb{R}^{k}$, where $\mathbf{y} = \mathbf{U}_{k}^{T}\mathbf{x}$
- Mapping the lower dimensional, transformed data back to the original host space and coordinate system: given the transformed data point $\mathbf{y} \in \mathbb{R}^{k}$, the representation of that point in the original host space and coordinate system is $\tilde{\mathbf{x}} = \mathbf{U}_{k}\mathbf{y}$.
- Projecting onto a lower dimensional subspace: given $\mathbf{x} \in \mathbb{R}^{n}$, PCA defines an orthogonal projector $\mathbf{P}_{k} = \mathbf{U}_{k}\mathbf{U}_{k}^{T}$, so that the projected data point $\tilde{\mathbf{x}} \in \mathbb{R}^{n}$ is defined by $\tilde{\mathbf{x}} = \mathbf{P}_{k}\mathbf{x}$. Projection is equal to the second mapping in this list composed with the first mapping in this list.
If you translate all of your data by a point $\mathbf{x}_{0} \in \mathbb{R}^{n}$, just replace $\mathbf{x}$ in the above list with the expression $(\mathbf{x} - \mathbf{x}_{0})$, and the same expressions will hold.
