Linear Algebra: Orthogonality and Projections

Explain the concept of Hilbert space in the context of quantum mechanics.

Let W be the subspace of $\mathbb{R}^2$ spanned by (1, 1). Find the orthogonal projection $P_1$ from $\mathbb{R}^2$ to W and the orthogonal projection $P_2$ from $\mathbb{R}^2$ to the orthogonal complement of W.

Verify that the orthogonal projections $P_1$ and $P_2$ satisfy the properties of orthogonal projection: $P^2 = P$, $P = P^T$, $P_1 + P_2 = I$, and $P_1P_2 = P_2P_1 = 0$.

Decompose (1, 0) along (1, 1) using the orthogonal projections $P_1$ and $P_2$.

Given a vector $\mathbf{u}$ and another nonzero vector $\mathbf{a}$ in $\mathbb{R}^3$, find the vector component of $\mathbf{u}$ along $\mathbf{a}$ and the vector component of $\mathbf{u}$ orthogonal to $\mathbf{a}$.

Let B be a set of vectors $v_1, v_2, \ldots, v_k$ such that all vectors in B have length 1 $(\|\|v_i\|\| = 1)$ for all $i$ and are orthogonal to each other $(v_i \cdot v_j = 0)$ for $i \ne j$. Show that B is an orthonormal set and prove that B is also linearly independent.

Given the vectors $v_1 = \left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$ and $v_2 = \left(\frac{2}{3}, \frac{1}{3}, -\frac{2}{3}\right)$, determine if they form an orthonormal set in $\mathbb{R}^3$.

Show that the given vectors form an orthogonal basis for $\mathbb{R}^3$. Then, express the given vector $\mathbf{w}$ as a linear combination of these basis vectors. Give the coordinates of vector $\mathbf{w}$ with respect to the orthogonal basis.