# 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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### Theorem regarding direct sums

Let $w_1$ and $w_2$ be subspaces of V. Prove that V is direct sum of $w_1$ an $w_2$ iff each vector in V can be uniquely written as $x_1 + x_2$ where $x_1$ belongs to $w_1$ and $x_2$ belongs to $w_2$ ...
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### Real and imaginary part of tensors of matrices

Given a matrix $A\in \mathbb{C}^{n\times m}$, clearly we can write $A=\Re(A)+i \Im(A)$, i.e., the real and imaginary part of $A$. (For instance, $A=[1,i]$, then $A=[1,0]+i[0,1]$). I am interested in ...
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### what is Expected Mean

Thus the expected mean $\mu$ of the set $\mathcal S$ can be given as \begin{align*} \mathbb E \mu&= \sigma^2+\frac 1r \sum_{i=1}^m\left(\mathbb E\lambda_i-\sigma^2\right)\\ &\geq \sigma^2+\...
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### Calculating the missing two points of rectangle if 2 points and the aspect ratio are known

How can I calculate the missing two points of a rectangle if I know 2 points (top left and top right) and the aspect ratio i.e 16:10. For example: Top left: A(834, 449) and Top right: B(1675, 423)
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### Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
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### Question on proof of number of solutions of linear system

The proof my book uses starts off by saying: "If the system has exactly one solution or no solutions, then there is nothing to prove", and then continues on by assuming there is an infinite ...
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### prove subspace of a vector space

Consider the subset $T$ of $\mathbb{R}^2$ defined as follows: $T := \left\{(x, y) : x, y \in \mathbb{R} : y = 3x \right\}$. Prove that T is a subspace of the vector space $\mathbb{R}^2$. My attempt: ...
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### Is there a unique projection map in this case?

Let $X$ be a Banach space over $\mathbb{C}$. Let $A,B$ be closed subspaces of $X$ such that $X=A\oplus B$. Assume that $||a+b||=||a||+||b||$ for each $(a,b)\in A\times B$. Then, does there exist a ...
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### How to rotate a coordinate system in $\mathbb{R}^3$ through an angle about an arbitrary axis passing through origin?

The question spurred in my mind when I was asked the following: Find the transformation matrix T that describes a rotation by $120^\circ$ about an axis from the origin through the point $(1,1,1)$....
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### Does the line pass through the origin? [on hold]

Does the line $L: (3,4,-1)+s(7,-2,1),\ s\in \Bbb R$ pass through the origin? I'm not sure how to do this.
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### Conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z}) × GL_m(\mathbb{Z}/q\mathbb{Z})$ [on hold]

Find the number of conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z})× GL_m(\mathbb{Z}/q\mathbb{Z})$ of cyclic subgroups of order pq?.
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### Determine the distance between the point $(2,-3,1)$ and the point of intersection of three planes

Determine the distance between the point $(2,-3,1)$ and the point of intersection of the following system. $3x-y+z=4$ $-x+2y+3z=7$ $x+3y+4z=12$ I'm not quite sure how to do this?
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### (Double) Coset of $GL(n, q^2)/GL(n, q)$

I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the ...
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### Geometry Of Unitary Transformations

Ever since I first took Linear Algebra, I have over time realized how concepts like determinants, eigenvalues, diagonalization, orthogonal transformations and so on have very intuitive geometric ...
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### Linear maps uniqueness proof: Difference between “uniquely determined on span($v_1,…,v_n$)” and “uniquely determined on V”

I've just been introduced to linear maps, and I'm still trying to wrap my head around the proofs, which don't read easily for me. In the uniqueness proof, the very last section states: Thus $T$ (the ...
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### Why doesn't transposing matter?

I was helping a friend with linear algebra, particularly I was teaching how to check if a collection of vectors $v_1, ..., v_k \in \mathbb{R}^n$ are linearly independent (assuming the vectors are ...
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### Intuitive understanding of the matrix of a linear transformation

Is it accurate to say that a matrix $M(T)$ of the linear map $T:V\to W$ encodes the linear map into a series of numbers by showing how the linear map applied to the basis vectors of $V$ can be ...
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### Number of multisets with restrictions on specific element count

I am looking to find the number of multisets with restrictions on the number of specific elements. This isn't for homework, it is a work related problem. My set of items is {A, a, B, b}. I want to ...
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### Why does all vectors achieve steady state after multiplying by Markov matrices for infinite number of times? [on hold]

what is so special about Markov matrices and how to prove this statement?
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### Linear Algebra Over Integer (Book Recommendation)

I am doing research on Integer Spline. So, there are lots of system of linear diophantine equations appear, which I have to solve them within the integer. I am lacking the theory behind the system of ...
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### Relationship between eigenvalues of Hermetian matrices

Suppose that we have two $m\times m$ matrices $A$ and $B$ which are Hermetian, with $|B_{ij}|\leq |A_{ij}|$ for $i,j = 1,2 \cdots, m$. Can we say anything about the relationship between the largest ...
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### Is there always $n$ permutations of a vector in $R^n$ that are linearly independent?

As long as the $n$ entries of the vector are all different and they dont add up to zero. If it is true, how to prove it, if not, what is a counter example?
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### How to find the determinant of this $n \times n$ matrix in a clever way?

\begin{bmatrix} b_1 & b_2 & b_3 & \cdots & b_{n-1} & 0 \\ a_1 & 0 & 0 & \cdots & 0 & b_1 \\ 0 & a_2 & 0 & \cdots & 0 &...
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### Good true-false linear algebra questions?

Can you suggest me a collection of true-false linear algebra questions, like the ones found in the MIT exams, if possible with solutions (i.e. explanations)? Sorry if it turns out that my request is ...
I have some questions about the uniqueness of matrices when post- and pre-multiplied with vectors (inner product). Say we have two vectors $\vec{a}$ and $\vec{b}$, whose inner product is a scalar, ...
While attempting to answer a question here (namely, the finite dimensional case of the title question: Prove that if $\lambda$ is an eigenvalue of $T$, a linear transformation whose matrix ...