# Questions tagged [linear-algebra]

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

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### Trying to understand Grobner basis

While studying Grobner basis, I realized that creating a basis from a given set of polynomials is not that hard, it is reduced to solving with Gauss Jordan a system of equations. What I don't ...
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### Direct sum of two subspaces - how to show

I haven't found this exact question yet on this site: If $U$ and $W$ are subspaces of the inner product space $(\mathbb{R}, V, +, \langle .,. \rangle)$, and I have to show that $U \oplus W = V$, ...
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### Positive-Definite Matrix Question

I want to prove that the matrix is positive definite using the fact that: If $A$ is symmetric and $\langle x, Ax \rangle$ > $0$ for a nonzero vector $x$ then $A$ is positive. So I have the ...
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### If $A$ is $n \times n$ non singular complex matrix and $B = (\bar A)' A$, where $(\bar A)'$ is the conjugate transpose of $A$ then…

If $A$ is $n \times n$ non singular complex matrix and $B = (\bar A)' A$, where $(\bar A)'$ is the conjugate transpose of $A$. If $x$ is an eigenvalue of $B$ then $x$ is real and positive. (True/false)...
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### Do $A$ and $A^T A$ share an eigenvector?

I have been learning about singular value decomposition from http://www.ams.org/publicoutreach/feature-column/fcarc-svd and they say that orthongoal vectors in the domain are mapped to orthogonal ...
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### (A−λI)x=0 and x≠0 iff det(A−λI)=0: Why [[1,1],[1,1]][[2],[3]] = [[5],[5]] ≠ 0 when det([[1,1],[1,1]]) = 0?

As refered to Why non-trivial solution only if determinant is zero, I wonder why \begin{gather} \begin{bmatrix} 1 & 1 \\1 & 1 \end{bmatrix} \begin{bmatrix} 2 \\3 \end{bmatrix} = \begin{...
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### Understanding formula for projection of a vector onto a line

The formula for projection of a vector 'b' on line represented by vector 'a' is given as the following in Linear algebra and its applications by Gilbert Strang But why is that after finding the ...
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### Given similar $A,B$ matrices how to find a non-singular matrix $P$ such that $B=P^{-1}AP$.

Given 2 similar square matrices $A$ and $B$ of same order,how to get hold of a non-singular matrix $P$ such that $B=P^{-1}AP$.One way I know is to solve the system $PB=AP$ which often becomes hard to ...
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### How to compute (unipotent) radicals

My question follows some previous one, essentially this one. I want to understand, given an algebraic group $G$ (say linear), how to compute its radical and unipotent radical. The (unipotent) radical ...
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### How to avoid “what goes up must come down” in motion equations

I am simulating a slot machine spinning and would like to start with an initial velocity and gradually apply a negative acceleration until the total displacement has reached the desired slot. The ...
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### Characterizing linear maps with matrices

Bosch Linear Algebra page 92 We want to prove that the map $\psi:\operatorname{Hom}_K(V,W)\rightarrow K^{m\times n}$ with $f\mapsto > A_{f,X,Y}$ is an isomorphism. I have not understood why we ...
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### Is it always possible to swap columns of a matrix by a left hand side multiplication?

I was thinking about swapping the columns of the matrix. It is well known that if you want to swap 2 columns of a matrix, you do a right hand side multiplication with a permutation matrix $T_{ij}$, ...
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### Prove equivalence of two statements related the vector field $V$

Let $M$ and $N$ be linear subspaces of $V$. Prove that (a) if $y\in M$, $z\in N$ and $y+z=\theta$ then $y=z=\theta$ is equivalent to (b) if $y+z = y^{\prime}+z^{\prime}$, where $y,y^{\prime}\in M$ ...
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### Singular Value Decomposition (SVD) - Odd step in proof

There are a lot of proofs about the singular value decomposition of a matrix $A \in \mathbb{R}^{m \times n}$. Now this is the start of the proof in my textbook: Take the symmetric matrix $A^T \cdot A$...
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### Parameterizing rotation matrix

A general rotation matrix in terms of Euler angles is given by $$\mathcal{R}=R_{\hat z}(\alpha)R_{\hat y}(\beta) R_{\hat z}(\gamma).$$ Working out the matrix multiplication we obtain the known ...
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### A System of Equalities and Inequalities; Is There A Unique Solution?

Suppose $m_x,M_x,m_y,M_y\in\mathbb{R}$ are known and $x_l,x_h,y_l,y_h\in\mathbb{R}$ are unknown. Does the simultaneous system \begin{align*} x_l \hspace{1mm}+\hspace{1mm} x_h &\hspace{1mm}=\...
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### Linear combinations of cash flows

We consider cash flow vectors over T time periods, with a positive entry meaning a payment received, and negative meaning a payment made. A (unit) single period loan, at time period t, is the T-vector ...
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### Non-trivial $\operatorname{Hom}_R(M;N)$ [on hold]

In K. Conrad's handout on dual modules(https://kconrad.math.uconn.edu/blurbs/linmultialg/dualmod.pdf), there is the following theorem-exercise 2.11, which states: Let $R$ be an integral domain with ...
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### Prove that for a bilinear symmetric form $B$ there's a vector v$≠$0 that B(v,v)=0 iff $B$ and $-B$ are not positive

a bit of a messy question: Suppose there's a Bilinear symmetric form $B$ on vector space $V$ above $R$. how can I prove that there's a $v ∈ V$ that sustains $B(v,v)=0$ iff $B$ and $-B$ are not ...
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### Geometrically speaking: what distinguishes a subspace vs a non-subspace in finite dimensional vector space?

Suppose we have $\mathbb{R}$ Then imagine an interval in $\mathbb{R}$, let's call it $I$. Then $I$ is not a subspace, since addition of two elements do not necessarily lie in the set. It seems to ...
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### How to find stationary values of $\frac{1}{2}x^TAx-x^tb$?

How to find stationary values of $\frac{1}{2}x^TAx-x^tb$? We know that we need to differentiate it with respect to $x$. But I dont know how to differentiate? Can someone help
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### Why we pick $0$ vector to check in linear indepenence?

I know that linear independence means each vector is not the linear combination of the others. But, I don't know why when we check whether a set of vectors are linearly independent, we only check for ...
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### For the purposes of DFT, is “the” primitive root of unity $w_n = e^{ 2\pi i / n }$ or $w_n = e^{-2\pi i / n }$?

I'm working on the section of Strang's Linear Algebra 4e that discusses discrete Fourier transforms. To make the exercises easier, I wrote myself a Python script that generates the $n$th Fourier ...
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### Proving a space is a subspace

I've been given a list of spaces and asked to see if they are subspaces R2 Here's one thats giving me trouble $$\begin{pmatrix} x\\ y\end{pmatrix}: x^2 = -y^2$$ I understand to prove its a valid ...
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### Probability density function after adding values to the set and transforming

Suppose I have matrix $\mathbf{N}$ (of size $m$ by $n$) of values that has the PDF $f(X)$. Then, I vertically concatenate $\mathbf{N}$ on top of a zero matrix of size $k$ by $n$, like this: \begin{...
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### How to find $\textsf T^t$ on $\mathbb R^2$, equipped with the standard inner product?

When given a linear operator $\textsf T: \mathbb R^2 \to \mathbb R^2$ defined by $$\textsf T(x,y) = (x+y,x+2y)$$ How can I find $\textsf T^t$ on the standard inner product space $\mathbb R^2$?
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### Is there a function that satisfies this property?

Let $n=(n_1,\dots,n_d)$ be a $d$ dimensional vector of positive integers in which $d$ is fixed but $n$ is not. Let $r=(r_1,\dots,r_d)$ be defined such that $n_i\,r_i=1$ for $i = 1,\dots,d$. Is there a ...
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### Method for expressing determinant as product of 2 determinants.Again is there any for matrices?

Is there any specific technique to write a determinant as a product of 2 determinants. I have so long kept it doing manually.At the same time I want to ask can any matrix be expressed as a product of ...
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### Linear algebra and matrix kernels

I have a set of vectors $v_i \in \mathbb{R}^n$, $i=1,...,m$ and I want to know if there exists a matrix $A \ne 0$ such that $v_i \in \operatorname{Ker}(A)$ for all $i$. Now my approach is to show ...
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### Construction of tensor algebra: extending the multiplication from being defined on pure tensors

T.S.Blyth in his book Module Theory: An Approach to Linear Algebra defined multiplication on the tensor algebra $\bigotimes M = \bigoplus_{n \in \mathbb{N}}\bigotimes^n M$ by first defining it for ...
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### Proving that a linear transformation is diagonalisable

Given that $V = \mathbb{R}[X]_{\leq 2}$ and $\alpha \in \mathbb{R}$ prove that the linear transformation $L: V \to V$ given by $L(P(X)) = \alpha P(X) + (X+1)P'(X)$ is diagonalisable and determine the ...
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