# Linear Algebra Summary

## The language of Quantum Mechanics

While we describe physical systems in a bunch of different ways, linear algebra is the best way to describe the behaviour of quantum objects. It's also the best way to translate the theory behind whats going on into practical protocols for developing quantum algorithms.

## Material

The purpose of this section is to touch on each of the most important topics in linear algebra, without going into too much detail. If you're reading through and find yourself unfamiliar with a particular topic, you'd benefit from taking some time to review it.

### Review Material

* Vectors
* Matrix operations
* Linear combinations
* Linear independence / dependence
* Vector spaces
* Basis and span
* Transformations

### New Material

* Bra-ket notation

## Resources

### Additional Info & Problem Sets Summary

Below is a list of all of the extra detail & problem sets included in this section of the tutorial:

#### [Basics](/learning-quantum/linear-algebra/basics.md)

* [An introduction to matrices](https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-intro-to-matrices/v/introduction-to-the-matrix)
* [Detailed Review of vectors](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/vector-introduction-linear-algebra)
* [Matrix addition and subtraction problems](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:adding-and-subtracting-matrices/e/matrix_addition_and_subtraction)
* [Scalar multiplication problems](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:multiplying-matrices-by-scalars/e/scalar_matrix_multiplication)
* [Matrix multiplication problems](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:multiplying-matrices-by-matrices/e/multiplying_a_matrix_by_a_matrix)

#### [Linear Operations](/learning-quantum/linear-algebra/vector-relationships.md)

* [Detailed explanation of linear independence](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-independence/v/linear-algebra-introduction-to-linear-independence)
* [Detailed explanation of linear combinations](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-combinations/v/linear-combinations-and-span)

#### [Vector Spaces](/learning-quantum/linear-algebra/space-dimension-and-span.md)

* [More about linear sub-spaces and span](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/subspace-basis/v/linear-subspaces)
* [Detailed description of basis](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/subspace-basis/v/linear-algebra-basis-of-a-subspace)

#### [Transformations](/learning-quantum/linear-algebra/transformations.md)

* [Fun transformation visualization tool](https://demonstrations.wolfram.com/MatrixTransformation/)
* [More information about transformations](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/transforming-position-vector)
* [Transformation matrix problems](https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-matrices-as-transformations/e/multiplying_a_matrix_by_a_vector)
* [More specific details about linear transformations and how they work](https://www.khanacademy.org/math/linear-algebra/matrix-transformations/linear-transformations/v/linear-transformations)


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