# Tagged Questions

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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### Finding the Orthogonal Complement to a subspace

So suppose I have a vector space, $V$ which is all continuous functions on $[0,1]$. Additionally, we have an inner product over $V$ where $\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$. Now suppose I ...
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### Relationship between geometric multiplicity, algebraic multiplicity and left and right eigenvectors of a matrix

The following statement is from the book Matrix Analysis by Horn and Johnson. An eigenvalue λ with geometric multiplicity 1 can have algebraic multiplicity 2 or more, but this can happen only if ...
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### Help solving the equation [on hold]

I'm stuck and don't know what to do next to solve this equation. Any hints? $y(x_2−x_1)−y_1(x_2−x_1)=x(y_2−y_1)−x_1(y_2−y_1)$
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### Which one is equation of tangent

Is equation of tangent plane $z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} )$ or $z=f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} )$ In my book I found ...
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### Properties of transfer matrices and their traces

I'm having difficulties understanding some arguments in my statistical mechanics lecture and would like to make them more rigorous by proving some properties. For the Ising model on a lattice we ...
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### Determine if the following vectors are coplanar.

I have no idea to start with this question, I know how to find if vectors are coplanar when the values of the vectors are given to me, but I do not know how to manipulate coplanarity properties well ...
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### Can one factor matrices?

I know that one can factor integers as a product of prime numbers. Is there an analog of it to matrices? Can we define prime matrices such that every matrix is a product of prime matrices? Is there ...
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### Area of the region bounded by four vectors.

I'm stuck on how to approach this problem. I have a feeling it involves determinants and linear algebra. It's to find the area of the region bounded by the vectors: [-7,7], [5,5], [3, -4], [-5,-6]
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### Linear equation [duplicate]

I have been troubles with the problem below, The line 4x-5y+20=0 cuts the x axis at A(0,4) and the y axis at B(-5,0). Find the equation of the median through O of triangle OAB. Find the equation of ...
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### Why do we use the word “scalar” and not “number” in Linear Algebra?

During a year and half of studying Linear Algebra in academy, I have never questioned why we use the word "scalar" and not "number". When I started the course our professor said we would use "scalar" ...
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### Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k$

A quadratics question. Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k.$ I recently asked a similar question, but this problem seems ...
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### the rational canonical

let T$\in$ $\mathcal{L}$($\mathbb{Q^3}$,$\mathbb{Q^3}$) be given by $$T(v)= \left[ \matrix { 1&-1&-4 \\ 1&-1&-3 \\ -1&2&-2 } \right]v$$ . Find the rational canonical form ...
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### How to normalize and inverse a vector so it sums to 1 ?

I understand how normalization works. You sum up the individual values of the vector, you divide each value by the sum, and voila... they sum to 1. Why doesn't it work when you subtract them from ...
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### Calculating inverse function with 2 variables

$f: R^{2}\mapsto R^{2}$ $(x,y)\mapsto (x^{2}-4y^{2}+x, -xy+3y)$ I should calculate inverse function of $f$ in point $(3,1)$. I tried to do $(x,y)\mapsto(u,v)$, but I just dont know how to get x ...
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### Points on which function is invertible?

$f: \mathbb R ^{2}\mapsto \mathbb R ^{2}$ $f(x,y)\mapsto((x-y)^{2}+1, x-y^{3}-2)$ For which points is this function invertible? I calculated the Jacobian matrix, but what should I do next to get ...
In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...