Linear Algebra Checklist

Linear algebra is the native finite-dimensional language of quantum mechanics. Even when a system is described by wavefunctions, the same ideas appear as inner products, bases, eigenvalue equations, unitary changes of representation, and tensor products.

This checklist tells you what to be able to do before relying on the formalism pages.

You Should Be Able To

• Work with complex vectors and matrices.
• Compute inner products using the physics convention that $\langle\phi\vert\psi\rangle$ is conjugate-linear in the first slot.
• Normalize a vector.
• Expand a vector in an orthonormal basis.
• Change basis with a unitary matrix.
• Compute eigenvalues and eigenvectors of small matrices.
• Recognize Hermitian, unitary, projection, and normal matrices.
• Use spectral decomposition for a finite-dimensional Hermitian matrix.
• Compute expectation values such as $\langle\psi\vert A\vert\psi\rangle$.
• Form tensor products of vectors and operators.
• Keep subsystem ordering fixed in a product basis.

Why This Matters in Quantum Mechanics

Quantum states are represented by vectors or density matrices. Observables are represented by Hermitian operators in the standard projective setting. Measurement probabilities come from projections and inner products. Time evolution in closed finite-dimensional systems is unitary. Composite systems use tensor products.

If the linear algebra is shaky, common conceptual mistakes become harder to catch: confusing a vector with its coordinate list, confusing a basis with a measurement, treating amplitudes as probabilities, or changing the order of tensor-product factors without noticing.

Minimum Examples

You should be comfortable with the following examples before entering the main formalism:

Task	Example
Normalize a state	$\lvert\psi\rangle=(1,i,2)^T/N$
Diagonalize an observable	$A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$
Change basis	move between $\{\lvert0\rangle,\lvert1\rangle\}$ and $\{\lvert+\rangle,\lvert-\rangle\}$
Use a projector	$P=\lvert0\rangle\langle0\rvert$
Form a tensor product	$\lvert0\rangle\otimes\lvert1\rangle=\lvert01\rangle$
Check unitarity	verify $U^\dagger U=I$

Diagnostic Problems

1. Normalize the vector $v=(1,i,2)^T$.

Solution

The norm squared is $v^\dagger v=1+1+4=6$, so the normalized vector is

$$\frac{1}{\sqrt6}(1,i,2)^T.$$

2. Find normalized eigenvectors of

$$\sigma_x= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}.$$

Solution

The eigenvalues are $+1$ and $-1$. Normalized eigenvectors may be chosen as

$$\lvert+\rangle=\frac{1}{\sqrt2} \begin{pmatrix}1\\1\end{pmatrix}, \qquad \lvert-\rangle=\frac{1}{\sqrt2} \begin{pmatrix}1\\-1\end{pmatrix}.$$

3. Let $\lvert\psi\rangle=(\lvert0\rangle+i\lvert1\rangle)/\sqrt2$. Compute the probability of the outcome associated with $P=\lvert0\rangle\langle0\rvert$.

Solution

The probability is

$$\langle\psi\vert P\vert\psi\rangle =\lvert\langle0\vert\psi\rangle\rvert^2 =\frac{1}{2}.$$

4. In the product basis ordered as $\{\lvert00\rangle,\lvert01\rangle,\lvert10\rangle,\lvert11\rangle\}$, which basis vector is $\lvert1\rangle\otimes\lvert0\rangle$?

Solution

With that ordering, $\lvert1\rangle\otimes\lvert0\rangle=\lvert10\rangle$, the third basis vector. Changing the subsystem ordering would change the coordinate placement, so the ordering convention must be declared.

Where to Review

Use these Toolkit pages when a checklist item is weak:

• Complex Vector Spaces
• Inner Products
• Matrices as Linear Maps
• Finite-Dimensional Hilbert Spaces
• Eigenvalues and Eigenvectors
• Hermitian Operators
• Unitary Operators
• Projectors
• Spectral Decomposition
• Tensor Products

References

• G. Strang, Linear Algebra and Its Applications, 4th ed., Brooks/Cole, 2006.
• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.