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.

Filter by
Sorted by
Tagged with
0
votes
0answers
9 views

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 ...
0
votes
0answers
19 views

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 $U \oplus W = V$, ...
0
votes
1answer
15 views

Distributing intersection over vector space addition.

Let $U, S, W$ be three subspaces of the vector space $V$. Prove or disprove the following a) $U ∩ (S+W) ⊆ (U ∩ S) + (U ∩ W)$ b) $(U ∩ S) + W ⊆ (U + W) ∩ (S + W)$ My Approach: Let $v ∈ U ∩ ...
0
votes
0answers
33 views

Question regarding the similarity of an invertible matrix with its inverse .

Find the set $S$ of all possible $n×n$ invertible matrices $M$ such that $M$ is similar to $M^{-1}$ . My approach Actually, I was thinking about this problem when I came across a theorem stating ...
0
votes
0answers
13 views

Generalized eigenvalue problem for non-singular matrix

I'd like to solve a generalized eigenvalue problem of the form $$Ax=\lambda Bx$$ However, matrix $B$ is full-rank, Hermitian, positive semidefinite and satisfies $\mbox{Tr}(B)=a$. Matrix $A$ is ...
1
vote
0answers
12 views

Are the zero eigenvalues of a Laplacian matrix semi-simple?

It is known that the Laplacian matrix $\mathcal{L}$ for a directed weighted graph has at least one zero eigenvalue. If it has more than one zero eigenvalue, will there be non-trivial Jordan blocks ...
2
votes
4answers
36 views

Solve System of equation using elimination?

\begin{align} I:&& ~~ x+\frac12y &= 6 \\[.5em] II:&& ~~ \frac32x + \frac{3}{2}y &= {17 \over 2} \end{align} when $x$ was multiplied by $(-3/2)$ in first equation the $x$ will ...
2
votes
1answer
28 views

Trace of matrix $A^{\ast}A$

Given a $n \times n$ matrix $A$ with complex entries. And $A^{\ast}$ represents the conjugate transpose of $A$.Then If $\left | tr{\left ( A^{\ast}A \right )}\right | <n^2$, then $\left |a_{...
0
votes
1answer
17 views

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 ...
0
votes
1answer
13 views

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)...
0
votes
1answer
58 views

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 ...
-3
votes
1answer
32 views

(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{...
0
votes
1answer
24 views

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 ...
2
votes
1answer
51 views

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 ...
0
votes
0answers
14 views

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 ...
0
votes
2answers
69 views

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 ...
1
vote
1answer
20 views

$(C_{1}\cup C_{2})^{\perp}=C_{1}^{\perp} \cap C_{2}^{\perp}$, linear code $C_{1}, C_{2}$

Prove $(C_{1} + C_{2})^{\perp}=C_{1}^{\perp} \cap C_{2}^{\perp}$ for any linear code $C_{1}, C_{2}$ over $\mathbb{F}_{q}$ of the same length. we know $C^{\perp}= \{ x \in \mathbb{F}_{q}^{n}: <x,v&...
1
vote
1answer
15 views

Showing the Matrix identity $A_{g\circ f,X,Z}=A_{g,Y,Z}\cdot A_{f,X,Y}$

We want to show the indentity in a specific way that I did not understand, the definitions and theorems which are stated in the proof will be listed. The book is Bosch Linear Algebra page 95, Th4: $...
1
vote
1answer
38 views

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 ...
0
votes
1answer
38 views

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 $T_{ij}$, ...
2
votes
1answer
32 views

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 $y,y^{\prime}\in M$ ...
0
votes
1answer
16 views

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 $A^T \cdot A$...
0
votes
0answers
7 views

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 ...
0
votes
1answer
27 views

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}=\...
0
votes
0answers
29 views

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 ...
2
votes
1answer
29 views

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 ...
1
vote
1answer
27 views

Formula for rotating vectors

Let $u = (u_1, u_2)$ and $x=(x,y)$ a rotation of u by an angle θ. Then $\left\lVert u \right\rVert = \left\lVert x \right\rVert$ We know: $$cosθ = {{u \cdot x}\over \left\lVert u \right\rVert \cdot \...
7
votes
1answer
57 views

How wide is the Birkhoff Polytope?

Define the width of a polytope $P \subset \mathbb R^d$ as the minimum length of the interval $\{v \cdot p:p \in P\}$ for $v$ in the unit sphere. In other words the width is the smallest number $W$ ...
3
votes
3answers
220 views

Showing that the limit of non-eigenvector goes to infinity

Let $A$ be a $3$ by $3$ real matrix with the triple eigenvalue $1$. Also, further suppose its eigenspace corresponding to $1$ is only of dimension $1$. Thus, we can find a basis of $\mathbb{R}^3$, ...
0
votes
1answer
19 views

definite positive symmetric bilinear form

Let A be the matrix of a positive definite symmetric bilinear form. Prove ִִ$a_1,$$_1$$a_n,$$_n$≥$a$$_1,$$_n$$a_n,$$_1$ I dont really have any idea what to do here.
0
votes
1answer
29 views

There exists a smallest linear subspace that contains every $M\in\mathcal{M}$

This is Exercise 7 from Berberian (1992) Linear Algebra, p.23. Let $V$ be a vector space and let $\mathcal{M}$ be any set of linear subspaces of $V$. Let A be the union of the subspaces in $\mathcal{...
0
votes
1answer
20 views

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 ...
1
vote
1answer
28 views

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 ...
0
votes
1answer
25 views

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
1
vote
3answers
86 views

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 ...
1
vote
3answers
51 views

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 ...
0
votes
1answer
37 views

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 ...
2
votes
0answers
59 views

Linear transformations that preserve a subspace

Let $V$ be an $n$-dimensional vector space (say over $Z_2$) and let $U$ be a subspace of dimension $k$. I was wondering if there is a way to characterize full-rank $n\times n$ linear transformations $...
-3
votes
1answer
35 views

find a particular solution of the indicated linear system that satisfies the initial conditions x1(0)=3 and x2(0)=-7 [on hold]

We have yet to do any examples like this in class so I am confused on what to do, I'm not even sure how to get it started. The results should be the particular solutions of x1(t) and x2(t). I attached ...
1
vote
2answers
24 views

Would a matrix have more than one reduced echelon matrix if each reduced echelon matrix was achieved using different elementary row operations?

I’m working on a problem asking me to find a general solution of a matrix with infinite solutions. The ordered pairs I’m getting are much different than what the book says. The book also says that “...
1
vote
1answer
35 views

Nullity of a linear transformation, Tom Apostol calculus II problem

$Tom M Apostol, Calculus II$ $Exercise$ $ 2.4 $ $Q$ $26)$ Let V be a the linear space of all real functions continuous on [a,b]. If $ f$ $\in $ V, $g$=$T(f)$ means that $ g(x)=\int_{a}^{b}\ f(t) ...
-2
votes
0answers
23 views

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{...
0
votes
2answers
64 views

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 $\mathbb R^2$?
1
vote
0answers
110 views

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 ...
1
vote
0answers
29 views

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 ...
0
votes
1answer
42 views

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 ...
2
votes
2answers
26 views

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 ...
2
votes
4answers
72 views

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 ...
1
vote
1answer
49 views

Eigenvalues of a sum of two matrices: $\mathbf{A}+\mathbf{D}$

Let $\mathbf{A}$ be a random matrix of size $N\times N$ and let $\mathbf{D}$ be any diagonal matrix. If we denote $\{\alpha_i\}$ the eigenvalues of $\mathbf{A}$ ($\alpha_i \in \mathbb{C}$) and $\{\...
2
votes
4answers
63 views

Matrixes of higher order like $M_{\aleph\times \aleph}$ [on hold]

I was wondering if thinking about matrixes of the form $M_{\aleph\times \aleph}$ and assigning properties to them like matrix multiplication makes sense and useful in some way. note: $\aleph$ is the ...