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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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1answer
25 views

Prove the set of functions $T$ is a vector space.

Show that $$T=\{t(x)|t(x)=a(x^2-1)+b\ln x+c\cot x\}$$ is a vector space. My attempt: To prove a vector space is to prove for $x_1,x_2\in T$, $x_1+x_2\in T$. So I calculated $$t(x_1)+t(x_2)=a({x_1}^...
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votes
1answer
43 views

Does this special identity matrix have a name?

recently I stumbled upon the problem of defining a diagonal matrix whose elements are identity matrices of $dim = n$, where $n$ is the column/row index. For example, for $n=3$: $\mathbb{I}_3 = \left[{...
0
votes
1answer
28 views

Matrix equation with several X

Solving matrix equation $A^2X-B^T = 3X$ (to find X), I'm trying to do next thing: $A^2X-B^T = 3X$ $A^2X-3X = B^T$ $(A^2-3)X = B^T$ Can we do it in that way and, if yes, what should we do with $(A^2 ...
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3answers
40 views

What does subspace of $R^n$ mean?

I'm having trouble visualizing what the subspace of $R^n$ looks like. I've been taught it means a "space" which is a subset of the larger space $R^n$, but then why is something like S = {[1, 1]} not ...
2
votes
1answer
37 views

Dimension of the space of homogeneous polynomials in a quotient ring

Consider the ring $k[x_1,x_2,x_3,x_4]/(x_1x_3,x_1x_4,x_2x_3,x_2x_4)$. How do I find the dimension of the vector space of homogeneous representatives of the quotient ring of each degree? I've only ...
1
vote
2answers
27 views

Linear Algebra - Proof of subspace

I'm stuck in how to proof the follow question. Let $V = \mathcal{F}({I},\mathbb{R}) $, where ${I}$ is in the range of $[0,1]$ and $$T = \left\{ \mathcal{f} \in V: f(1) = 0 \right\} $$ Is $T$ a ...
0
votes
1answer
20 views

Can you find the z component in this system?

I will use uppercase letters to denote known variables and lowercase letters to denote unkown ones. Assume I have a vector $<X', Y', z'>$, and an invertible matrix $T$, and a vector $<x,y,z&...
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1answer
15 views

The Effect of Adding an Edge on the Laplacian of a Weighted Digraph

Let $G$ be a weighted digraph with Laplacian $L:=D-A$, where $D$ is the degree matrix and $A$ is the incidence matrix. Is there any result on the behavior of the eigenvalues of $L$ when we add an edge ...
3
votes
2answers
62 views

Dimension of vector spaces with finitely many elements

Background - I am currently taking an introductory linear algebra course and we were introduced to vector spaces. Now, at first it seemed pretty much 'obvious' that the only vector space with ...
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1answer
34 views

Linear algebra problem in the final exam [on hold]

We all know in a linear transformation, dim(V)=dim(ker(T))+dim(range(T)). Prove that there exist no linear transformation R5->R2 such: Ker(T)={x1,x2,x3,x4,x5/x1=2(x2),x3=2(x4)=x5}
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0answers
26 views

minimise the trace of a matrix over all column permutations

I have a 10x10 positive symmetric matrix, I need to find the optimal permutation of the columns in order to minimise the trace. I can't try all permutations because that would be a 10! problem. Any ...
3
votes
1answer
45 views

Combinatorics problem on k sets and partionning a set of numbers, understanding of the proof

The question I'm struggling with is given here : Remove any number and the remaining numbers can be partitioned into two subsets of equal sum; prove all numbers are equal. The problem is the ...
0
votes
1answer
23 views

Exponent operations: addition, multiplication, grouping etc.

I get easily confused while evaluating exponents: when should I add them and multiply them? We add exponents when multiplying two numbers are of the same base: i.e., $ (2^2) . (2^3) = 2^5 $ - When ...
0
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1answer
25 views

Find $\text{card}(T_n( \mathbb R ) \cap O(n))$.

Let $T_n( \mathbb R )$ be the set of upper triangular matrices of size $n$. Let $O(n)$ be the set of general orthogonal matrices and $SO(n)$ the set of special orthogonal matrices. Find the cardinal ...
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2answers
23 views

why is $ \{(0,x,z)|x,z\in R\}$ is a two dimensional subspace space of $R^{3}$ over $R$ but $\{(0,0,z)|z\in R\} $ $\cup$ $\{(0,x,0)|x\in R\}$ is not?

why is $ A=\{(0,x,z)|x,z\in R\}$ is a two dimensional subspace space of $R^{3}$ over $R$ but $B=\{(0,0,z)|z\in R\} $ $\cup$ $\{(0,x,0)|x\in R\}$ is not? i Think both are two dimensional as A has ...
2
votes
3answers
38 views

Find the area of parallelogram and its missing vertex

Given three radius-vectors: $OA(5; 1; 4), OB(6;2;3), OC(4;2;4)$, find the missing vertex $D$ and calculate the area of obtained parallelogram. My attempt: Firstly, we are to find the vectors which ...
-4
votes
1answer
26 views

If $\omega$ is non real cube roots of unity , then the eigenvalues of the matrix

If $\omega$ is a non-real cube root of unity, then what are the eigenvalues of the matrix $$\left[ \begin{matrix} 1 & 1 & 1\\ 1 & \omega & \omega^{2} \\ 1 & \omega^{2} & ...
1
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0answers
24 views

Linear operator cancellation

I came across this by looking at Green's function motivation topic: suppose that $L$ is a linear operator (acting on a function, distribution (as in this case) or whathever). Take the equation: $$L \...
0
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1answer
27 views

Endomorphism on polynomial vector space $\mathbb{R}_3[x]$

Let $ \mathbb{R}_3[x] $ be the vector space of the polynomials with the degree $ \le 3$. Given the endomorphism on this vector space, $$ T:\mathbb{R}_3[x] \to \mathbb{R}_3[x], T(f)(x) = f(x+1)-f(x), $$...
5
votes
1answer
51 views

Prove that $\langle\mathbf{A}, \mathbf{C}\rangle \leq \delta$ equals with $\|\mathbf{A}\|_*\leq\delta$

Given an arbitrary matrix $\mathbf{A}\in R^{n\times n}$ and the basis matrix set $\mathbb{S}=\{\mathbf{C}\in R^{n\times n}: \mathbf{C}^T\mathbf{C}=\mathbf{I}_n\}$. Then how to prove: 1:If we have $\...
0
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1answer
56 views

$ \langle A, B\rangle := \operatorname{Tr}(AB^{∗}) $ Inner Product in $\mathbb{C} $?

We define $\langle\cdot, \cdot\rangle $: $\mathbb{C}^{n\times n} \times \mathbb{C}^{n\times n}\longrightarrow \mathbb{C} $ by $ \langle A, B\rangle := \operatorname{Tr}(AB^{∗})$. Show that this ...
1
vote
1answer
13 views

Linear independence proof of sublist from a list of dependent vectors

Let $\lambda$ be an eigenvalue of $A$, such that no eigenvector of $A$ associated with $\lambda$ has a zero entry. Then prove that every list of $n-1$ columns of $A-\lambda I$ is linearly independent. ...
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votes
2answers
23 views

If A is a nxn singular matrix, then it has a singular value = 0

This is a question on a testexam. But am I correct in assuming that a singular matrix has det = 0, which gives it an eigenvalue of 0 and that gives it a singular value of 0?
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0answers
30 views

Let $E=[1,2,…n]$ where $n$ is an odd integer.

Let $E=[1,2,...n]$ where $n$ is an odd integer. Let $V:$ vector space of all functions from $E$ to $\mathbb{R}^3$ such that $(f+g)(k)=f(k)+g(k)$ and $(\lambda f)(k)=\lambda f(k)$ where $k \in E$ and $...
0
votes
1answer
21 views

matrix algebra and idempotent matrix

I'm having a little trouble understanding a few derivations in my book for least squares regression. $\textbf{Question 1}$: If $\textbf{M}^0 \textbf{i} = [\textbf{I} - \frac{1}{n}\textbf{ii'}]\...
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votes
0answers
21 views

Example on Field [on hold]

as we know there is a bit difference in the definition of different Fields. Can anyone suggest me some concrete examples differentiating Field, Sigma Field, and Borel Field?
2
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2answers
53 views

How to quickly tell if a set is linearly independent or not?

Before proving if a set is a basis for $R^n$, I have to determine if the set is linearly independent or not. We aren't allowed to use matrices, and I want to save time during a quiz. What are some ...
0
votes
1answer
16 views

Matlab - Matrix Projections

For homework, I would like to compute the projection of the first row of $ B $ ( $ B $ is an undefined large matrix) onto the third row of $ B $. Here is what I have tried text = 'dot(B(1,:),B(3,:...
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votes
1answer
18 views

Find the distance between a point and given parametric line?

Find the distance of the point $(3,6,−5)$ from the line $r(t)=\langle 1+3t,1+4t,5−3t\rangle$.
2
votes
2answers
61 views

About a problem of linear transformation $ ST-TS=I $

The problem is, $S$ & $T$ are two linear operator i.e. belongs to $$L(V)$$ such that $ST-TS=I_n$ then prove that $L(V)$ is infinite dimensional vector space. Now, I have showed that for all $n\in ...
0
votes
1answer
37 views

$c\lambda$ is an eigenvalue of $cA$?

"Let $A$ a square matrix have an eigenvalue $\lambda$ with corresponding eigenvector $v$. Let $c\in F$. Then $c\lambda$ is an eigenvalue of $cA$." This is true right? since the eigenvalues of $A$ ...
0
votes
1answer
21 views

algebraic expression of Matrix product [on hold]

Suppose $M = X^T \Delta X$, where $X$ and $\Delta$ are $P \times P$ matrices and $\Delta$ is symmetric. Can anyone give a simple algebraic expression of the matrix $M$?
6
votes
2answers
64 views

Showing $ 2 $ matrices are similar [duplicate]

I gotta show if or if not those $ 2 $ matrices are similar: $$ \left(\begin{matrix} 3 & 2 & -2 \\ 1 & 4 & 0 \\ -2 & 1 & -1 \\ \end{matrix}\right) $$ $$ ...
2
votes
1answer
34 views

How is it true? if a polynomial has no solutions then it is irreducable.

So I found this statement in my friend's notes and I think it's wrong for example if $x^2 + 1 = 0$ has no solution and is not reducable but if I square the whole thing then it is reducable but still ...
0
votes
1answer
14 views

Eigenvalues of a block off-diagonal matrix

Let $A_1,A_2 \in \mathbb{R}^{n \times n}$. Construct the block matrix $A$ as follows: $$A: = \left[ {\begin{array}{*{20}{c}} 0&{{A_1}}\\ {{A_2}}&0 \end{array}} \right]$$ My observation is that ...
0
votes
1answer
12 views

When product of matrix and vector equals 0, why does product of matrix, it's transpose and vector also equals 0?

I've got a matrix A and a vector s When Av = 0, why does A*As = 0 and also s*A*As = 0? A* is transposed matrix A.
3
votes
1answer
23 views

Find rank of AB, given that A has linearly independent columns and B has rank 2

I'm trying to prove to myself that given... Matrix A, which has linearly independent columns, and at least 2 columns... Matrix B, which has rank of 2 Their product, AB, will have rank of 2. I ...
2
votes
1answer
39 views

If union of n subspaces of V is a subspace of V, then one of the n subspaces must contain the other n-1 subspaces

Help prove this? I can prove for n=2, but I'm stuck on proving it for general n. Thanks! My proof for n=2 Forward direction: Consider A and B and A $\cup$ B is a subspace of V. We prove by ...
1
vote
1answer
26 views

Finding ch. polynomial, min. polynomial and the Jordan n. form of $f$ knowing $f^3=0$ and $f(v_1)=f(v_2)=v_3, f(v_3)=kv_4, f(v_4)\in<v_1,v_4>$

Given a vector space $V$ of dimension $4$ and a base $\{v_1,v_2,v_3,v_4\}$, let $f$ be an endomorphism of $V$ such that $f^3=0$ and $f(v_1)=f(v_2)=v_3, f(v_3)=kv_4, f(v_4)\in<v_1,v_4>$, where $k$...
1
vote
2answers
17 views

Procedure to find a pair of matrices whose product has rank less than each matrix in the pair

I am trying to figure out how to find two rank-deficient matrices (not necessarily square) which when multiplied will have rank less than either of the original matrices. In other words I am looking ...
0
votes
2answers
40 views

Linear operator statement

I'm stuck with a practice problem question: '' Is it true that a linear operator, T : V → V, on one-dimensional vector space V over field F has the form T(v) = av, for all v ∈ V and a a scalar from ...
0
votes
1answer
40 views

Parametric equations intuition

If I paramaterise a function like x= f(t) and y=g(t) like x=t and y=t^2 why why does eliminating the parameter give you the function that the parametric equations? It may be clear for y=x^2 but i'm ...
0
votes
2answers
24 views

transition from one base to another

Given the polynomials $p_1,p_2,p_3,v_1,v_2,v_3 \in P_2$: \begin{gather*} p_1(t) := t^2 − 2t + 5, \qquad p_2(t) := 2t^2 − 3t, \qquad p_3(t) := t + 1, \\ v_1(t) := t^2 + 4t − 3, \qquad v_2(t) := t − 1,...
1
vote
1answer
22 views

How to express a quadric equation from canonical form to a different basis.

I have the quadric $3X^2-Y^2-Z^2=0$ expressed in the canonical form, and the matrix of change of basis from a basis B to the canonical form is $$P=\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\...
1
vote
1answer
23 views

Locally trivial line bundle

I am reading the book "An Introduction to Contact Topology" by Geiges. In the proof of Lemma 1.1 where he proves that any co-dimension 1 hyperplane field distribution is locally kernel of a 1-form, ...
0
votes
1answer
17 views

If $B$ is a basis of $V$ and $U\subseteq V$ is linearly independent then there exists a $C\subseteq B$ such that$ \tilde{B}:= U\dot\cup C$ is a basis

It is an application of Zorn's Lemma, It would help me a lot if somebody could explain one part of the proof that I did not understand Why can we describe every element of $B$ as a linear ...
0
votes
0answers
13 views

Is working out the smith normal form trivial?

What's the point of using elementary column operations to work out the smith normal form of a matrix of reals when it is easy to work out exactly how many redundant equations there are from the row ...
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votes
1answer
26 views

Why is $c'c=1$ a normalisation to solve for eigenvector $c$? Why is it imposed? [on hold]

When finding the eigenvalues/eigenvectors for matrix $A$
0
votes
0answers
33 views

How to prove the limit $\lim_{k\rightarrow\infty}\lambda^{k-j}\binom{k}{k-j}=0,\; k>j,$ and $k\gt j$

I find difficulty proving the following limit: Suppose $|\lambda|<1$,then $$ \lim_{k\rightarrow\infty}\lambda^{k-j}\binom{k}{k-j}=0,\quad k>j. $$ When I try to prove that the $k$th Jordan ...
2
votes
2answers
78 views

Subspace of a vector space problem

So the task says: for $\alpha_1\ldots,\alpha_n, \beta\in \mathbb R$, we define $U = \{(x_1\ldots,x_n) ∈ \mathbb{R}^n \,|\, \sum_{i=1}^n \alpha_ix_i = \beta\}$. When is the $U$ subspace? I know that ...