# Questions tagged [linear-algebra]

For questions about 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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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### For which alpha does this system of equations spiral towards the orgin?

I am stuck with a math problem and I hope that somebody could help me in the right direction. Question: For which values of $\alpha$ do the trajectories of the solutions of the systems $x^{\prime}=A x$...
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### Basis of polynomial vector space with conditions

I understand that the monomial basis proposed in this answer: $\{1,x,x^2,x^3,\ldots,x^n\}$ spans a regular polynomial vector space, but what process would I use to create a basis when there is ...
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### How do i generate cubic coordinate-space traversal functions?

Goal- generation of a function f(x) such that for each x = 0 to infinite, f(x) outputs a point in a coordinate space ...
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### Why do two distinct hyperplanes in $\mathbb{R}^4$ intersect in a plane?

Any tips? Ideas? A proof would be helpful
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### Lower bound on the off-diagonal elements of a PSD matrix

Suppose we have a PSD matrix $X\in\mathbb{R}^{2d}$, which could be written in the following block form $$X=[X_1\quad X_2;\quad X_2^\top\quad X_3],$$ where $X_1, X_3\in\mathbb{R}^d$ are PSD matrices, ...
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### Linear Transformation from $\mathbb{R}^6$ to $P_2(\mathbb{R})$

I am trying to figure out what the transformation of the values {$x_1, x_2, x_3, x_4, x_5, x_6$} $\in \mathbb{R}^6$, becomes when transformed to $P_2(\mathbb{R})$. I understand that $P_2(\mathbb{R})$ ...
1 vote
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### Cauchy-Schwarz inequality and angle between two vectors

Notes I am reading these notes, and I can't understand the Cauchy-Schwarz inequality. It says that it proves that the input is between $[-1,1]$. The Cauchy-Schwarz inequality only states that the ...
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### Multiplication of two random matrices over a finite field

Consider a matrix $\mathrm{X}$ sampled uniformly at random from the set of all rank $r$ matrices over $\mathbb{F}_q^{m \times n}$ and a matrix $\mathrm{Y}$ sampled uniformly at random from the set of ...
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### Given a subspace $W$ of $V$ and a linearly independent set $A \subseteq W$, is $A$ linearly independent in $V$?

Can you please show that, if a set $A$ is linearly independent in $W$, a subspace of $V$, then it is linearly independent in $V$? Thank you. Here is my attempt. Let $A = \{ a_1, a_2, \cdots , a_n \}$ ...
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### Given two points, construct a parallelepiped with a square base(length given)

So I have been doing this for a while now, and I am afraid I am horribly overthinking this problem, so I came here for some fresh takes on this. I want to obtain all the vertices of a parallelepiped ...
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### What is the formula for the orbital velocity of the Earth from x, y, z coordinates of ephemerides at set time intervals?

For example for step size 10080 minutes x y z v 1721057.5 B.C. 0001-Jan-01 -5.83E-01 7.93E-01 3.65E-03 1721064.5 B.C. 0001-Jan-08 -6.78E-01 7.16E-...
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### Add some constraints to make $x^TAy \geq 0$ where $A \succ 0$

I am trying to prove the difference of two algorithms' asymptotic mean square error. Eventually, I get an expression like $x^TAy$, where $A$ is a positive definite matrix. I would love to see under ...
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### (Dis)prove for a matrix $A$ and for an unitary matrix $S$, where $D = S^{-1}AS$ a diagonal is which diagonalelements are imaginary that $A^{*} = - A$

$A \in M_{n,n}(\mathbb{C})$ a complex matrix and $S \in U_{n}$ an unitary matrix with $D = S^{-1}AS$ a diagonalmatrix, which has only imaginary elements in it's diagonal. (the real part elements are ...
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### Intersection of linear manifold

I have this question on my homework sheet that I don't know how to approach: "A three-dimensional linear manifold in R4 has the direction vectors a1=<2,0,1,0> a2=<1,1,0,0> a3=<0,1,...
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### maximal eigenvalue of self-adjoint operator is non-degenerate

I want some help in this one, if someone can prove or disprove it: "If $T$ is a compact, self-adjoint operator with positive spectral gap, then $||T||_2$ is always an eigenvalue and the ...
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### Exercise 3.F.5 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 3.94 Definition dual space, $V'$ The dual space of $V$, denoted $V'$, is the vector space of all linear functionals on ...
### What is the relation between matrices $A^2$, $A^{-1}$, $A^2 + 2A + I$ to the diagonal matrix $D$ corresponding to $A$? [closed]
Find the eigenvalues and eigenvectors of $A^2$, $A^{-1}$, $A^2 + 2A + I$. Write the relation that connects these matrices to the diagonal matrix D corresponding to A. Can you use this relation to ...