Cocalc 1 Intro To Linear Algebra Ipynb

Linear algebra, the study of vectors, vector spaces, and linear transformations, is fundamental to modern mathematics and its applications. From solving systems of equations (known to ancient Chinese and Babylonian mathematicians) to quantum mechanics and machine learning, linear algebra provides essential computational and theoretical tools. This notebook explores core linear algebra concepts using SageMath, demonstrating matrix operations, eigenvalue computations, decompositions, and applications to solving linear systems. A vector is an ordered list of numbers, while a matrix is a rectangular array. Matrices can represent linear transformations, systems of equations, and data tables. The modern notation for matrices was introduced by Arthur Cayley in 1858, though determinants and systems of linear equations were studied much earlier by mathematicians like Leibniz and Cramer in the 17th-18th centuries.

Matrix transpose swaps rows and columns, denoted ATA^TAT. A matrix is symmetric if A=ATA = A^TA=AT and skew-symmetric if A=−ATA = -A^TA=−AT. There was an error while loading. Please reload this page. There was an error while loading. Please reload this page.

Linear algebra is the language of quantum computing. It is therefore crucial to develop a good understanding of the basic mathematical concepts that linear algebra is built upon, in order to arrive at many of the amazing and interesting constructions seen in... The goal of this section is to create a foundation of introductory linear algebra knowledge, upon which the reader can build during their study of quantum computing. We will start our investigation into introductory linear algebra by first discussing one of the most important mathematical quantities in quantum computation: the vector. Formally, a vector ∣v⟩|v\rangle∣v⟩ is defined as elements of a set known as a vector space. A more intuitive and geometric definition is that a vector "is a mathematical quantity with both direction and magnitude".

For instance, consider a vector with xxx and yyy components of the form (35)\begin{pmatrix} 3 \\ 5 \end{pmatrix}(35​). This vector can be visualized as an arrow pointing in the direction of 333 units along the xxx axis and 555 units along the yyy axis: Note that "tail" of the vector doesn't have to be positioned at the origin; it only needs to point in the correct direction. In quantum computing, we often deal with state vectors, which are simply vectors that point to a specific point in space that corresponds to a particular quantum state. This can be visualized using a Bloch sphere. For instance, a vector representing the state of a quantum system could look something like this arrow, enclosed inside the Bloch sphere, which is the so-called "state space" of all possible points to which...

There was an error while loading. Please reload this page. Author: Andreas Noack Jensen (MIT) (http://www.econ.ku.dk/phdstudent/noack/) (with edits from Jane Herriman) Notice that AAA has type Array{Int64,2} but xxx has type Array{Float64,1}. Julia defines the aliases Vector{Type}=Array{Type,1} and Matrix{Type}=Array{Type,2}. Many of the basic operations are the same as in other languages

As in other languages A' is the conjugate transpose, or adjoint The problem Ax=bAx=bAx=b for square AAA is solved by the \ function.

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