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Linear Algebra Review

Relationship between linear algebra and quantum computing

Quantum computing uses linear algebra throughout, and for nearly everything. Doing quantum computing without linear algebra would be like taking a course on Shakespeare without knowing English.

The parts of standard linear algebra we'll most use are:

Vectors, matrices, matrix-vector multiplication, matrix-matrix multiplication.
Span, basis, linear independence.
Eigenvectors and eigenvalues.

The new parts of linear algebra (that we'll cover after the review) involve:

A new type of notation called the Dirac (or ket) notation.
Vectors and matrices that have complex numbers in them.
A few new concepts such as: outer-products, projectors, and special types of matrices called Hermitian and unitary matrices.

What is likely to be a bit confusing:

The new (Dirac) notation takes some getting used to.
If you haven't seen complex numbers before, they need a little practice.
Unfortunately, unlike the arrows we've used for regular real-valued vectors, complex vectors do not have a convenient visualization. Thus, you'll need to get used to symbols and abstraction.

Accordingly:

To avoid the double whammy of new notation and concepts simultaneously, we'll first review in standard notation and then review again in the new notation.
The only way to get used to the new concepts is to develop facility through practice problems.

Review of basic linear algebra

Please review in this order:

Review Part-I: Vectors, dot-products, lengths, angles, orthogonality, matrix-vector multiplication, solving \({\bf Ax} = {\bf b}\) exactly and approximately.

Review Part-II: matrix-matrix multiplication, spaces, span, basis, independence, orthogonality (again), projections.

Review Part-III: summary of useful basic results, summary of some key theorems.

Review Part-IV: change of basis for vectors and matrices

The above review sections are from the linear algebra course.

In the remainder, we will selectively review other topics, covering only those aspects relevant to quantum computing.

If you have not completed the above 4-part review by now, please do so before continuing here.

Some new linear algebra

The linear-algebra review parts listed above (parts I - IV) are topics you have very likely studied in a prior linear algebra course.

The next section is really a review but some new linear algebra, that extends what you have seen before:

Part-V: Some new linear algebra concepts

Back to main review page