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21

I was a teaching assistant in Linear Algebra previous semester and I collected a few applications to present to my students. This is one of them: Google's PageRank algorithm This algorithm is the "heart" of the search engine and sorts documents of the world-wide-web by their "importance" in decreasing order. For the sake of simplicity, let us look at a ...


15

Another very useful application of Linear algebra is Image Compression (Using the SVD) Any real matrix $A$ can be written as $$A = U \Sigma V^T = \sum_{i=1}^{\operatorname{rank}(A)} u_i \sigma_i v_i^T,$$ where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal matrix. Every greyscale image can be represented as a matrix of the intensity ...


9

This is a simpler example, but maybe that'll be good for undergraduate students: Linear algebra is a central tool in 3-d graphics. If you want to use a computer to represent, say, a spaceship spinning around in space, then you take the initial vertices of the space ship in $\mathbb{R}^3$ and hit them by some rotation matrix every $.02$ seconds or so. Then ...


7

We can also use Linear Algebra to solve Ordinary Differential Equations An ODE is of the form $$\underline{u}'(t) = A \underline{u}(t) + \underline{b}(t)$$ with $A \in \mathbb{C}^{n \times n}$ and $\underline{b}(t) \in \mathbb{C}^{n \times 1}$. If we have an initial condition $$\underline{u}(t_0) = \underline{u_0}$$ this is an initial value problem. ...


5

The restricted isometry property (RIP) of matrices is something not too hard for undergraduates to understand: it means that a (rectangular) matrix $A$ satisfies $$ (1-\delta) \|x\| \le \|Ax\|\le (1+\delta)\|x\| \tag{1}$$ for all vectors $x$ with at most $s$ nonzero components. The constant $\delta$ should be small, and of course independent of $x$. The ...


5

Using the identity $$\sin x=\cos\left(x-\frac\pi2\right)$$ we see that the given matrix is a matrix of rotation of angle $\frac{4\pi }{9}$ hence the answer is $9$.


4

Some answers/hints: Yes the dimension of $GL(n,\mathbb{R})$ is $n^2$. In fact it's the open subset of $\mathbb{R}^{n^2}$ on which the determinant (which is continuous) is nonzero. The special linear group is the subset of the general linear group on which the determinant is one. Neither of these is itself a vector space, so talking about basis vectors ...


4

Multiplication of a graph's adjacency matrix can be used to calculate the number of walks of length $n$ from one vertex to another. In particular: Proposition. For any graph formed of vertices connected by edges, the number of possible walks of length $n$ from vertex $V_i$ to vertex $V_j$ is given by the $i,j^\text{th}$ entry of $A^n$, where $A$ is the ...


4

Stoichiometry (not that our students would ever stoop to linear algebra for it) is a very elementary place it shows up. Quantum mechanics is an advanced place. Linear programming is ubiquitous in business applications, as is game theory in economics and political science, and a lot of game theory is based on linear algebra and Markov processes. Least squares ...


3

I worked as a software engineer for 27 years for a large Defense corporation. They used Finite Element software tools to model spacecraft designs for stress tests, amount of construction material required, simulated launch testing, etc. Finite Element theory is based on a matrix of vectors that describe the connections and forces on elements of a structure. ...


3

One of the examples my students have absolutely loved in the past is Hill cipher. It is a "real" application although outdated but the students do love playing with it. Using some sort of a encoding, convert a message to a bunch of numbers, and then a key matrix is chosen. The message can be organized into an array and multiplied by the key for encryption. ...


3

A closed subspace of a complete space is complete. All you need to do is verify that $V$ is a closed subspace of $C[0,1]$. But $\|f_n - f\|_\infty \to 0$ implies $f_n \to f$ pointwise. So if $f_n \in V$ converges to a function $f \in C[0,1]$ then $f_n(0) \to f(0)$ and $f_n(1) \to f(1)$. Thus $f \in V$ and $V$ is closed.


2

I like the Google Page Rank and Adjacency Matrix points. Linear Algebra is a deep subject that is readily connected to computer science, graph theory, and combinatorics in unexpected ways. The traditional connection is with numerical analysis. However, Linear Algebra is closely related to graph theory. There is a field known as algebraic graph theory which ...


2

Using Vandermonde matrices, one can show that for any $k$ and any $n$, there exists $k$ points in general position in $\Bbb R^n$. Indeed, given $k$ (assuming $k>n$), pick $k$ distinct real numbers and consider ${\bf v}_i=(r_i,r_i^2,\ldots,r_i^{n})$ for $i=1,\ldots,k$ and use Vandermonde's determinant to prove the claim.


2

Anything to do with scheduling and maximising linear systems: an airline scheduling planes and pilots to minimise the time airplanes are stationary and pilots are just sitting around waiting is an example. Linear optimisation saves millions if not billions of dollars each year by allowing companies to allocate resources optimally, and it's basically an ...


2

$u\in U + W$ does not mean that $u\in U$ or $w\in W$; rather, it means that $u$ is the sum of something in $U$ and something in $W$.


2

Hint: there is an obvious bijection from the unit circle to the interval $[0,2\pi)$ using angles.


2

By cofactor expansion along the fourth column, if you take the determinant $$\begin{vmatrix} 1&1&1&1\\ a&b&c&t\\ a^2&b^2&c^2&t^2\\ a^3&b^3&c^3&t^3 \end{vmatrix}$$ and find the coefficient of $t^2$, it will be the negative of the determinant you're after. This makes the problem a matter of understanding the ...


2

Can you describe in words what $T$ does to the components of a vector? If you can do this, then you can answer the first two questions easily. If you know that statement 2 holds, then for statement 3, if $k \ge n$, then $T^k=T^{k-n} T^n = T^{k-n} 0 = 0.$


2

Let $f(\mathbf v_E) = \mathbf v_B$. Can you show that the map $f$ satisfies the definition of a linear map, i.e. that $f(\mathbf u + \mathbf v) = f(\mathbf u) + f(\mathbf v)$ and that $f(\alpha\,\mathbf u) = \alpha\, f(\mathbf u)$? (Hint: You should be able to do this just by looking at the definition of $\mathbf v_B$.) OK, let me unpack that a little: ...


2

Here's a hint (assuming you are indeed missing a minus sign): $\sin \frac{\pi}{18} = \cos \frac{4\pi}{9}$.


1

Method: Express the equations in terms of $xy$, $x-y$ and then change variables.


1

Create a matrix $M$ whose columns are the $b_{i}$, then $T(u)=Mu$ performs the transformation. It is now easy to show that $T$ has the properties of a linear transformation.


1

Let $P(a,b,c)$ be the determinant in question. Then the general properties of determinants imply that: $P$ is a homogeneous polynomial of degree $4$. $P$ is alternating: that is, $P(a,b,c)=-P(b,a,c)=-P(a,c,b)=-P(c,b,a)$. In particular, $P$ vanishes whenever any two of $a,b,c$ are equal, and so it must have the linear homogeneous polynomials $(b-a)$, ...


1

Yes, it is correct. Uniform convergence implies pointwise convergence.


1

I think it might be easier to figure out what the complement is first. Notice that it's equal to $\left\{(x,y)\in\mathbb{R}^2|(\forall n\in\mathbb{N})\ y<x^{2n} \right\}$. Look at which (x,y) pairs belong to this set in cases when $|x|>1$ and $|x|\le 1$.


1

The matrix $$ A = \frac{1}{\sqrt 3} \left( \begin{array}{ll} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha \\ \end{array} \right) $$ is unitary. This is an instance of the more general Fourier Matrix, which answers your question for unitary matrices of arbitrary size. Note that these matrices are unitary and ...


1

Hint: Can you think of a matrix that transforms the first vector into the second? Then that's enough, since $T(v):=Av$ is a linear transformation.


1

One approach: Show that $T$ satisfies the definition of a linear transformation Another approach: Find the matrix $A$ satisfying $$ T(x) = Ax $$ For vectors $x \in \Bbb R^3$.


1

We have $\delta_i=\beta_i-\gamma_i$. It remains to prove that the family $(\delta_i)$ is linearly independent. Let $a_i\in\Bbb R$ such that $$\sum_i a_i\delta_i=0\iff \sum_ia_i\beta_i=\sum_i a_i\gamma_i$$ hence since $U\cap Z=\{0\}$ and $(\beta_i)$ is a basis for $U$ then $$\sum_i a_i\beta_i=0\implies a_i=0,\;\forall i$$



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