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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Conditioning in regards to matrix vector product

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
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1answer
8 views

Independance, Inverse and Additive Identity for vector with defined vector addition and scalar multiplication

I am struggling to find my bearings on this question. I am confident that I can do parts a and d. I have no clue how to approach b. I was also wondering if the redefined vector addition and scalar ...
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0answers
21 views

About a particular definition of “tensor”

I came across this quiet new to me way of defining "tensors", That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n ...
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1answer
21 views

How to tell that $W$ is a subspace of $ \mathbb R^3$?

To do this problem, I wrote this matrix in RREF form and found that $V_3$ is $-1V_1 + 2V_2$. This demonstrates that these planes are a basis for $ \mathbb R^2$. However, I am not sure to extend that ...
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1answer
19 views

normal operator equation

let $S: V \to V$ linear transformation in a inner product space of $\mathbb{C}$. Prove that $S$ is normal iff $$\|S(x)\| = \|S^*(x)\|$$ That's what I have done so far: if $S$ is normal than $$SS^* ...
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1answer
31 views

Prove that elementary matrices perform row operations

How to prove that elementary matrices actually perform their intended row operations: multiplying by a constant, adding a multiple of one row to another, and switching two rows? I've seen examples ...
3
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1answer
23 views

$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
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2answers
24 views

What is the meaning of the notation [A|B] in Linear Algebra.

I am going through Linear Algebra right now, we are using the book Elementary Linear Algebra by Andrilli. In one of the theorems he uses this notation without really introducing it. Here is the ...
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1answer
16 views

If $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$

Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then: if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary. if $A$ is normal and $f_A(x)=f_B(x)$ ...
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1answer
11 views

Write $F$ as a linear combination of elements of $\mathcal B^*$

If $V=\mathbb R[x]_k=\{\sum\limits_{i=1}^ka_ix^i:a_i\in\mathbb R, \forall i\}$ is a vector space of dimension $k+1$ over $K=\mathbb R$ and $\mathcal B=\{1,x,\dots,x^k\}$ is a basis of $V$. The dual ...
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7 views

How to change a basis of some orthogonal operator to obtain canonical form of operator [on hold]

We have a linear operator $f:X\rightarrow X$ on 3-dimensional real inner product space $X$ which has in O.N. basis $e_1, e_2, e_3$ a matrix $$ \left [ \begin{array}{rrr} \cos t & \sin t & 0\\ ...
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27 views

Linear Algebra - Transformations, image, kernel [on hold]

Question Define T : R3 → R3 by Tx = (x · (1, 0, −1))(1, 0, −1) + (x · (1, 1, 1))(1, 1, 1) (a) Compute the action of T on the unit vectors i, j, k. (b) Write down the standard ...
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29 views

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R\{1}. [on hold]

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R \ {1}. (a) Show that a ∗ b ∈ G for all a, b ∈ G. (b) Show that G together with the binary operation G × G → G, (a, b) |→ a ...
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2answers
30 views

The formula $\DeclareMathOperator{tr}{tr}\mathrm{adj}(A)=\tfrac{1}{2}[(\tr A)^2-\tr(A^2)]I_3-[\tr A]A+A^2$ for the adjoint of a $3\times 3$ matrix

Let $A$ be a square matrix of order $3$. Prove that $$\operatorname{adj}(A)=\frac{1}{2}[(\operatorname{tr} A)^2-\operatorname{tr}(A^2)]I_3-[\operatorname{tr} A]A+A^2$$ where $\operatorname{tr}A$ is ...
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22 views

Describing all the linear transformations satisfying the constraints

How to find the linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satistfying $4x_1-3x_2+x_3=0$ is a) Null space of $T$ b) Range of $T$ I'm not able to ...
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21 views

Linear Algebra - Transition matrices

Question I have some methodological questions with this exercise: 1. You are given that the transition matric $P_{\mathcal C,\mathcal B}$ from a basis $\mathcal B=\{b_1,\ b_2,\ b_3\}$ to a basis ...
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23 views

Examine if the set is linearly independent

How do I prove or disprove if $\{1, \cos x, \cos 2x,..., \cos nx\}$ is linearly independent? I tried solving the problem using the definition of linear independence, $\sum_{k=0}^n a_k\cos kx = 0$ ...
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0answers
21 views

Awkwardly formed linear spaces exercise

I came across such an exercise: Let $V$ be a linear space over $K$ such that $\dim V = n$. Show that for any $\alpha_1, \alpha_2, \dots, \alpha_m$ with $ m > n + 1$ there exist $a_1, \dots, ...
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4answers
54 views

How to find unknowns $w_1,w_2,w_3$ that satisfy $t=w_1f_1 + w_2f_2 + w_3f_3$?

For any $i \in \{1,2,3\}$, let: $w_i \in [0,1]$ is an unknown number such that $\sum_{i \in \{1,2,3\}} w_i = 1$. $t$ is a known number in $[0,1]$. Suppose that $t = 0.8$. $f_i$ is also a known ...
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2answers
31 views

how to determine a matrix has a single eigenvalue

Find the jordan form of the matrix $$A = \begin{pmatrix} 1 & 1 & 2 & 2\\ 1 & -2 & -1 & -1\\ -2 & 1 & -1 & -1\\ 1 & 1 & 2 & 2 \\ \end{pmatrix}$$ ...
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1answer
26 views

Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
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1answer
43 views

Linearity and invertibility of $A^{-1}$

If $A\in L(X)$ then prove that $A^{-1}$ is linear and invertible. Proof: Since $A$ is invertible then $A$ is injective and surjective. We know that $A^{-1}$ defines by $A^{-1}(Ax)=x$. Remark: Also ...
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1answer
21 views

Symmetric Matrix with Positive Eigenvalues

Not all matrix with positive eigenvalues is positive definite, i.e. $\mathbf{x}^\mathsf{T}A\mathbf{x}>0$ for all non zero vector $\mathbf{x}$. For example consider matrix $$A = \begin{bmatrix} 1 ...
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1answer
32 views

Show that $B^TAB$ is symmetric. [on hold]

$A$ is invertible, but it does not say that $A$ is symmetric. By $B^T$ I mean that $B$ is transposed.
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1answer
25 views

Is the the statement is true or false? [on hold]

Suppose $A$ is a $m \times n$ matrix and $V$ is a $m \times 1$ matrix with both $A$ and $V$ having rational entries and suppose the system $AX=V$ has a solution in $\mathbb{R}^n$. Then the equation ...
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1answer
17 views

If $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$

In one book on differential equations and dynamical systems I read that if (1) $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then (2) $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$. ...
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1answer
30 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
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1answer
22 views

$U,W$ are subspaces. show $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap W\}=\{U,W\}$

This is a question from a review package that is causing me some trouble. Let $U,W$ be subspaces of a finite dimensional vector space. Show if $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap ...
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1answer
20 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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2answers
32 views

Finding a Matrix B by knowing its Kernel is the Image of Matrix A

I understand how to find the image($A$). The basis of Im($A$) would be the first two columns of the matrix $A$ (given the two leading 1's in ref are in the first and second columns). So the ...
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2answers
29 views

Understanding a basic matrix theorem

There's a theorem in Linear Algebra which says that if ${\bf A}$ is an $m \times n$ matrix and $m < n$, then the homogeneous system of linear equations ${\bf A}{\bf X}=0$ has a non trivial ...
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3answers
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Showing that span$\{x,Ix\}$ is an invarient subspace of $V:=\mathbb{R}^n$

Let $V := R^n$ be a vector space and let $I \in O(n)$ be an operator satisfying $I^2 = -Id$. I want to show that the $span\{x,Ix\}$ is an invarient subspace of $I$. Let $W = span\{x,Ix\}$. I need to ...
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1answer
21 views

Proving a basis spans $R^3$

Doing some reviewing and I'm not 100% sure if my thought-process is correct. I have the following two vectors and need to prove they're a basis for $R^3$: $$B= \begin{bmatrix} 1 \\ ...
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1answer
16 views

Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space. Prove $\exists \{f_1,\ldots,f_n\}: (e_i,f_j)=\delta_{ij}$

Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space $V$. Prove there exists vectors $\{f_1,\ldots,f_n\}$ such that $(e_i,f_j)=\delta_{ij}$. I tried using ...
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0answers
42 views

Is $\{(x,y,z) \in \mathbb{R}^3 : x^2+3y^2+12z^2=0\}$ a vector space?

Is $\{(x,y,z) \in \mathbb{R}^3 :x^2+3y^2+12z^2 = 0\}$ a vector space? My inclination is that the only real solution to $x^2+3y^2+12z^2=0$ is $(0,0,0)$, which is the trivial subspace of ...
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1answer
14 views

Least Squares Solution and Singular Vector

Is there a simple way to show that the least square solution of an overdetermined linear system is equal to the right singular vector of the coefficient matrix corresponding to the smallest singular ...
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1answer
16 views

How can we know if the minimal polynomial of a matrix has no multiple products?

If the characteristic polynomial $f_A(x)$ has multiples of the same product, for example $f_A(x)= (x+2)^2(x-1)$ so $(x+2)$ has a multiple of $2$, then is there a condition on $A$ such that we know ...
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Why is the standard inner product on F^n equal to this?

In the textbook that I'm using the standard inner product is defined as $$\langle x,y\rangle = \sum_{i=1}^{n}a_{i}\overline{b_{i}}$$ where $x=(a_{1}, a_{2}, {...}, a_{n})$ and $y=(b_{1}, b_{2}, ...
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1answer
12 views

What is the dimension of $c_0/M$?

Let $c_0=\{ (x_n) : x_n\in \Bbb{R}, x_n \to 0\}$ and $M=\{(x_n)\in c_0 : x_1+x_2+\cdots + x_{10}=0\}$. Then, what is dim($c_0/M$) ?
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1answer
29 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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2answers
30 views

How to determine that the 3 points given in homogeneous coordinates are collinear? [on hold]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
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21 views

What is the maximum value of coefficient $f_v$ with the constraInt that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
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1answer
21 views

Sparse Matrix or Dense Matrix

My task is to implement the inner product and vector triad forms for a dense $A$ in single and double precision. I have successfully implemented the inner product and vector triad form although, I am ...
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15 views

PGL_n transformation fixing a set of n+2 points in general position must be identity

I was wondering if anybody can think of a slick and short argument to show that any $PGL_n(k)$ transformation $A$ that fixes a set of $n+2$ points $p_1, p_2,\dots, p_{n+2}$ in general position must be ...
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1answer
10 views

On a hypothetical computer with a word length of three digits and truncation, compute the solution of a system of equations

On a hypothetical computer with a word length of three digits and truncation, compute the solution of $$ \begin{matrix} -3x & + & y & = & -2 \\ 10x ...
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21 views

Set of conjugate vectors that span both Krylov space

If $P$ contains a set of conjugate vectors that span Krylov space of matrix $A$, i.e. $\mathcal{K}(A, x)$, and also $P$ span Krylov space of matrix $\mathcal{K}(B, x)$, is it true that the diagonal ...
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2answers
33 views

Find the matrix $P$ that multiplies $(x, y, z)$ to give $(y, z, x)$. Find the matrix $Q$ that multiplies $(y, z, x)$ to bring back $(x, y, z)$.

How do I solve these types of problems? What method or technique do I use? I want to learn how to solve the first one so that I can try to figure out the second part to the question myself.
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1answer
33 views

symplectic base in $\mathbb{R}^{2n}$

Please, can somebody help me? In the vectorial space $\mathbb{R}^{2n}$,is the canonical basis a symplectic one?
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2answers
28 views

Characteristic polynomial of $A^2$, given the characteristic polynomial of $A$

Let $A \in M^{\mathbb{R}}_{3x3}$, it's characteristic polynomial is $P_A(t) = t^3+t^2+t-3$. find the coefficient of the characteristic polynomial of $A^2$. I tried to solve it by finding the factors ...
2
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2answers
26 views

$A$ is diagonalizable, if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable

Given $A_{n\times n},B_{n\times n} \in \mathbb R$ such that $A$ is diagonalizable then: if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable over $ \mathbb R$. if $A,B$ ...