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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Proving an equality between two sums

Please, help me to prove that $\sum_{k=1}^m a_{kj}\left(\sum_{i=1}^n b_{ik}w_i\right) = \sum_{i=1}^m\left(\sum_{k=1}^n b_{ik}a_{kj}\right)w_i$
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31 views

Why can Echelon Matrices have zero rows but Echelon systems can't have any equations with no leading variables?

According to the definition my professor gave us its okay for a matrix in echelon form to have a zero row, but a system of equations in echelon form cannot have an equation with no leading variable. ...
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3answers
700 views

Is the inverse operation on matrices distributive?

For example, is the following true: $$(A + B)^{-1} = A^{-1} + B^{-1}$$ If $\det(A) \ne 0$, $\det(B) \ne 0$, and $\det(A + B) \ne 0$.
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1k views

Kernel and Image of a linear transformation T given on a basis

Let V be a 4-d vector space. $T:V \rightarrow V $is a linear operator whose effect on basis {$e_1, e_2,e_3,e_4$} is $Te_1= 2e_1- e_4$ $Te_2= -2e_1 + e_4$ $Te_3= -2e_1 + e_4$ $Te_4= e_1$ Find a ...
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1answer
29 views

Determining linear indepenence

I am having trouble with the dozens and dozens of rules for determining dependence, independence and generating sets and consistent and inconsistent. I know that these are all very closely related ...
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22 views

Transforming two dimensional into one dimensional binary summation

Consider two sets of real numbers $(v_i)_{i=1}^n$ and $(u_i)_{i=1}^n$. I want to find a sequence of real numbers $(t_i)_{i=1}^M$ such that $$ ...
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2answers
39 views

Stuck on finding rank of $T$ when $n=8$ of $T(A) = A - A^T$

$T: M_{n\times n}(F) \to M_{n\times n}(F)$ is a linear transformation. I know from rank-nullity that $\text{rank}(T) + \text{nullity}(T) = \dim(M_{n\times n}(F))$. I'm trying to find $N(T)$ and then ...
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54 views

Calculating the determinant of matrix.

The task is to calculate the determinant of following $n\times n$ matrix $A$: $a_{ij}=2$ if $i=j$. $a_{ij}=1$ if $|i−j|=1$ and $a_{ij}=0$ otherwise. I think the $\det A = n+1$. I got that result ...
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2answers
105 views

nilpotent and linear transformation

if $S$ is an $3$ by $3$ matrix with entries from real numbers why it's not possible that $S^4=0$ but $S^3$ is not zero? I'm thinking it's something related to nilpotent. And in nilpotent, there's a ...
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1answer
54 views

Recovering a dependent column vector from a matrix after row reducing that matrix

This is an old exam question I remember how to do, but not why it makes sense. Take a matrix $A$: $A=\left(\begin{array}{rrrr} 1 & 6 & 2 & -4 \\ -3 & 2 & -2 & -8 \\ 4 & ...
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68 views

Given a vector x, can we say something about an A such that A x = x?

Let us assume that a vector $x \in \mathbb{R}^n$ is given and we are looking for a matrix $A \in \mathbb{R}^{n\times n}$ which yields $A x = x$. That is, we perform a sort of reverse questioning: ...
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2answers
106 views

is the difference of a positive semi-definite matrix and its rank-1 approximation still positive semi-definite?

suppose $M$ is a symmetric positive semi-definite matrix, its largest entry occurs along the diagonal, say, is $m_{ii}$. Then we define vector $A$ as the column containing $m_{ii}$, i.e. (in Python ...
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542 views

Dot product of the column vectors from a matrix and their transposes through matrix multiplication

I have a matrix with data, every dataset is a column vector in my matrix. I want to know the dot product of the transpose of each column vector with the original column vector. If I transpose the ...
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1answer
108 views

Consider a game in which two players take turns removing any positive number of pebbles they want from one of two piles of pebbles.

Consider a game in which two players take turns removing any positive number of pebbles they want from one of two piles of pebbles. The player who removes the last pebble wins the game. Show that if ...
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1answer
37 views

Existence of a scalar

Suppose I have a vector $u\in\mathbb{R}^s$ with strict positive entries. Furthermore there is a vector $q\in\mathbb{R}^n$ and a matrix $A\in\mathbb{R}^{n\times s}$. I know that for any ...
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3answers
126 views

Sum of each row is $m$, prove that $m$ divides the determinant

This is a question from our reviewer for our exam for linear algebra. I just want to have some ideas how to tackle the problem. If $A$ is an $n\times n$ matrix with integer coefficients, such that ...
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0answers
51 views

Reference request on pseudo-determinants

I am looking for a reference on pseudo-determinants$^{(1)}$. I am mostly interested on general and/or basic equalities and properties such as those obtained for determinants. Any pointers would be ...
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2answers
50 views

Write a short proof for the following statement: Let n be a positive integer. Statement 1. If$ (\forall x\in \mathbb N, n|x)$, then n=1

Write a short proof for the following statement: Let n be a positive integer. Statement 1. If $(\forall x\in \mathbb N, n|x)$, then $n=1$. The proof I tried to explain is that since for every ...
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1answer
51 views

For every integer $a$, if $32 \nmid ((a^2 + 3)(a^2 + 7))$ then $a$ is even

Statement 1: for every integer $a$, if $32 \nmid ((a^2 + 3)(a^2 + 7))$ then $a$ is even I just want to double check my work because I'm not sure of my answers but the question asks, a) Rewrite the ...
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51 views

how to find thet a given real symmetric matrix is positive definite, positive semidefinite, negative definite, negative semidefinite or indefinite.

How to find that a given real symmetric matrix is positive definite, positive semidefinite, negative definite, negative semidefinite or indefinite.; on the basis of principle diagonal minor.
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95 views

Similar linear operators and change of coordinates

Let $S, T$ be operators in $\mathcal{L}(V)$, the space of all linear maps from $V$ to itself. In my lecture notes, I have the definition of similar: "We say that operators $S,T \in \mathcal{L}(V)$ ...
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169 views

Check if a Point is on Arc

I have a point with x,y,z co-ordinates. I have an arc,Its bulge, center point, midpoint,start point, end point,start angle and end angle. All i need to check is if the point exist on that arc or ...
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6k views

How to express a vector as a linear combination of others?

I have 3 vectors, $(0,3,1,-1), (6,0,5,1), (4,-7,1,3)$, and using Gaussian elimination I found that they are linearly dependent. The next question is to express each vector as a linear combination of ...
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1answer
81 views

Relating Two Inner Products Without Orthogonal Basis

Suppose the finite dimensional (real or complex) vector space $V$ has two inner products $\langle ,\rangle_1$ and $\langle,\rangle_2$. If we choose a basis, $\{v_1,\ldots,v_n\}$, each of these inner ...
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42 views

I need help on manipulating this expression:

Assume that $(4k + 3) ^ 2 - (4k + 3)$ is not divisible by 4. If this is true, prove that $(4(k+1) + 3) ^ 2 - (4(k+1) + 3)$ is not divisible by 4. I need to prove this for my induction problem, and ...
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1answer
119 views

Determining if a set is a generating set for $R^n$

I have \begin{bmatrix} -1 & 0 & 3 & -5 \\[0.3em] 1 & -1 & -7 & 7 \\[0.3em] 2 & 2 & 2 & 6 \end{bmatrix} and I need to prove if ...
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2answers
254 views

How to denote the opposite case of the Kronecker Delta?

The Kronecker delta is defined as link to wikipedia: $$\delta_{l,m} = \begin{cases} 1 & \text{if }m=l,\\ 0 & \text{if }m\neq l. \end{cases}$$ I would like to denote the case where: $$ = ...
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2answers
372 views

Vector space over complex numbers is also a vector space over the real numbers

So I know I need to prove the kajillion axioms of vector space like commutativity, associativity, the additive/multiplicative identities/inverses etc. How would I go about getting started?
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96 views

Multi objective optimization into single objective.

I read that it is possible to convert a multi-objective optimization problem into single objective by using weighted sum method. I wanted to know if it is a good idea to convert a two objective ...
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1answer
55 views

Subspaces of a vector

Prove that if $U$ and $W$ are subspaces of vector space $V$ such that $V=U\cup W$ then either $V=U$ or $V=W$. I am thinking one should use bases for $U$ and for $W$ somehow to show that bases for $V$ ...
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1answer
53 views

Diagonalizability of a linear transformation

I'm trying to prove the following: Let $ T: V -> V $ be a linear transformation and let $ \lambda_1 , ... , \lambda_k $ be distinct eigenvalues of T. Suppose the characteristic polynomial of T is ...
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48 views

About Linear Independence

If a vector in the set of vectors $$\{v_1,v_2,v_3...v_n\}$$ can be written as a linear combination of other vectors, must a row in the matrix formed by linearly combining the vectors be equal to a ...
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1answer
3k views

How do you determine whether a given set of functions is a subspace of C[-1,1]?

I'm having a terrible time understanding subspaces (and, well, linear algebra in general). I'm presented with the problem: Determine whether the following are subspaces of C[-1,1]: a) The set ...
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4answers
61 views

Can an $m \times n$ rank $1$ matrix be written as a product of an $m\times1$ and a $1\times n$ matrix?

If a $m \times n$ matrix has rank $1$, does it imply that it can be written as a product of one $m\times1$ and one $1\times n$ matrix. How to prove it ? Is this decomposition unique ? What are the ...
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1answer
43 views

Simplifying $\mathbf{X}(\mathbf{X}+a\mathbf{I})^{-1}$

I'm having trouble simplifying the following expression in matrix form: $$\mathbf{X}(\mathbf{X}+a\mathbf{I})^{-1}$$ Where $\mathbf{X}$ is an invertible $n \times n$ matrix, $a$ is a scalar value, ...
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45 views

Describe the subspace $T^{-1}(N)$

I got a) and b), but I have no idea about c).
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225 views

Is matrix multiplication by an invertible matrix one-to-one and onto?

Maybe I'm just not very experienced on the nitty gritties of matrix multiplication, but is the function $f(X)=AX$ where $X$ is a square matrix and $A$ is an invertible matrix one-to-one and onto? How ...
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1answer
123 views

How to prove $AB$ is a diagonalizable matrix?

Let $A$ be a positive definite matrix, $B$ an Hermitian matrix. How to prove $AB$ is a diagonalizable matrix?
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143 views

Is it possible to use the imaginary components of quaternions to facilitate calculation of vector cross products?

It has come to my attention that the cross products of the vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are almost identical to the products of the imaginary components of quaternions $i$, ...
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0answers
26 views

If $B_A\cup B_B$ is a basis for $E$, then not necessarely $E=A\oplus B$.

Consider the vector space $\mathbb R^3$ and an endomorphism $f:\mathbb R^3\rightarrow \mathbb R^3$. Suppose we are given $A$ the matrix of $f$ relative to the canonical bases, and from which we ...
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201 views

Quadratic form as a ratio of determinants

I am looking for some hints to prove the following equality: $y^{\top}y - y^{\top}X(X^{\top}X)^{-1}X^{\top}y = \dfrac{\det(L^{\top}L)}{\det(X^{\top}X)},$ where $y$ is a $n\times 1$ vector, $X$ is a ...
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1answer
44 views

Properties for internal stability of a discrete-time system

These are two parts of a larger proof I'm working on, can't figure how i) implies ii) though. Dynamic system: $x_{(k+1)} = Ax_{k}, x(0)=x_0$ Where $A \in \mathbb{R}^{n\times n} $ is a real ...
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2answers
811 views

Characteristic polynomial of an inverse

Given the characteristic polynomial $\chi_A$ of an invertible matrix $A$, I'm to find $\chi_{A^{-1}}$. I can see that this is theoretically possible. $\chi_A$ uniquely determines the similarity class ...
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85 views

What is the geometric meaning of the number of independent derivatives of $\gamma$?

Let $\gamma:I \to \mathbb{R}^n$ be a curve. I want to see, what is a geometric meaning of the number of independent derivatives of $\gamma$. I guessed it is it's dimension but it was not. Can you help ...
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203 views

If $A^2\succ B^2$, then necessarily $A\succ B$

I remember reading somewhere about the following properties of non-negative definite matrix. But I don't know how to prove it now. Let $A$ and $B$ be two non-negative definite matrices. If $A^2\succ ...
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1answer
55 views

can we decompose $\mathcal{A}$ of $\Bbb{R}^n$as an orthogonal transformation and a dilation?

Problem Any linear transformation of $\Bbb{R}^n$ is the composition of an orthogonal transformation and a dilation along perpendicular directions(with distinct coefficients) for any linear ...
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94 views

Direct sum and subspaces

The question states, "Find subspaces W, X, Y ⊂ ℝ2 with ℝ2=W⊕X=W⊕Y, but X does not equal Y." So if W and X are the Axis in R2 and Y can not equal X, how do you get W and Y to equal R2 without Y ...
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101 views

Transfomation of one coordinate system to a another

I have a molecule with one coordinate system ( denote as x,y,z ) where the origin is center of mass of the molecule. I have to define another coordinate system (p,q,r) for a local motion. (shown in ...
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6answers
2k views

Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...
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1answer
291 views

Subspaces of finite fields viewed as vector spaces on itself

How can I find the number of linear subspaces of dimensions 1 and 2 of the n- dimensional vector space $\mathbb{Z}^n_p$ over the field $\mathbb{Z}_p$?