Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Prove: the sum of simultaneously diagonalizable transformations is diagonalizable

Let $T, S$, linear transformations which are simultaneously diagonalizable. Prove that $T+S$ is diagonalizable. I need to rely on the the definition: $T,S$ are called simultaneously ...
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3answers
37 views

orthogonal and special orthogonal group of dimension $2$, group of isometries of $S_1$, $\mathbb{R}^2$ [on hold]

In my abstract algebra class, my teacher gave us this problem as to help review for the final. Unfortunately, I am not very well versed with linear algebra so I don't understand all that well what ...
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2answers
22 views

Proof that the kernel of an endomorphism to the power $n$ is a subset of the kernel of the endomorphism to the power $n+1$

I am expected to know how to prove the following but I can't seem to draw it out. Knowing that V is a Vector Space$$ T:V\to V $$ Prove the following $$ Ker(T^n)\subseteq Ker(T^{n+1}) $$ How ...
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2answers
40 views

Eigenvalues of a unimodular matrix

Let $U$ be a unimodular matrix, i.e. $U \in \mathbb{Z}^{n \times n}$, and $\text{det}(U) = \pm 1$. Do the real (or complex for that matter) eigenvalues of $U$ admit a special structure? Edit: It is ...
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2answers
56 views

Find Jordan form of a $3\times 3$ matrix

$$\left( \begin{array}{ccc} 0 & 1 & 2 \\ -5 &-3 & -7 \\ 1 & 0 & 0 \end{array} \right) $$ I figured out the eigenvalues are all -1 from the characteristic polynomial, but I'm ...
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0answers
20 views

interpertation of the determinant of an X'X product matrix (D-optimal design application)

As the title suggests, I have been looking into an application of D-optimal design. I read this thread What does it mean to have a determinant equal to zero? and found some of the answers ...
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1answer
30 views

Finding eigenvvalue and eigenspace

I am given a matrix $A= \bigg({} \matrix{10 & 7 \\-14 &-11} \bigg{)}$ and eigenvalue $3$. My elite mission is to find the treacherous basis for the eigenspace. I used the $(A -eI)=v$ where ...
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1answer
21 views

Using trigonometric identity to compute an inner product through an integral to form an orthogonal sequence of functions.

Consider the inner product space: $(C(0,L),\langle \cdot,\cdot \rangle),$ where: $\langle f,g \rangle = \displaystyle\int_0^L f(x)g(x)\, dx$. Use the trigonometric identity: $\sin(u)\sin(v) = ...
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2answers
32 views

Theorem with seemingly reduntant part

I encountered the following theorem in a linear algebra book: For any vectors $u, v$ in $R^n$ and any scalar $k$ in $R$: $u . u \geq 0$, and $u . u = 0 \iff u = 0$ I found the theorem in almost the ...
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23 views

Can I just use the following notation when proving a set is a vector space?

If given all functions of form $$f(x) = a + b \cos(x) + c \sin(x),$$ where $a,b,c$ are real numbers, would it be sufficient to use the notation "$f(x)$," when proving that the axioms hold and that ...
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2answers
28 views

Question about diagonalization and projections

Let a finite dimensional vector space $V$ above $\mathbb{F}$. Let $T:V\to V$ a diagonlizable transformation. We denote $a_1 \ldots a_r$ the $r$ different eigenvalues of $T$. By diagonalization, we ...
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1answer
36 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
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1answer
15 views

Projection of vectors

Compute $:$ $proj_\vec y (\vec x)$ $\vec{x}_1=\begin{bmatrix} 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}, \vec{y}_2 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix}$ Since the projection would be $:$ $(-2/0) * ...
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1answer
25 views

Solving linear differential equations system

Upon trying to solve this particular system , I've encountered a few problems. $$ y'=5y+4z $$ $$z'=-4y-3z$$ After solving for eigenvalues the quadratic yielded a double root at $\lambda=1$ . But I ...
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2answers
46 views

This answer is confusing $4\times 4$ eigenvalue calculation

Question: Find the rank and the four eigenvalues of the following matrix: $\begin{bmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & ...
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1answer
72 views

Two questions about diagonalization

Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
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1answer
19 views

Applying Gram-Schmidt process to a set of vectors to find first three polynomials orthogonal with respect to inner product

$.$ $\langle f, g \rangle = \displaystyle\int_{-1}^{1} f(x)g(x)dx$ Apply Gram-Shmidt process to the set of vectors $:$ {1, x, $x^2$, ...} to find the first three polynomials orthogonal with respect ...
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2answers
35 views

How to prove that a set is a vector space

How does one, formally, prove that something is a vector space. Take the following classic example: set of all functions of form $f(x) = á_0 + a_1x + a_2x^2$, where $a_i \in \mathbb{R}$. Prove ...
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1answer
28 views

Quadratic form - vector/matrix

I have two very simple (stupid) questions about quadratic forms. Having any matrices $A,B$ and vectors $x,y$ (real/complex, singular/regular, rectangular, infinite size, etc.) with appropriate size ...
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25 views

Inner prodcut space for complex numbers including complex conjugation

$..$ Consider inner product space : $(C, \langle \cdot,\cdot\rangle)$: where for complex numbers $..$ $\langle z_1, z_2 \rangle = \sqrt(z_1 *\overline{z_2}$) Computing $..$ $\langle 2-3i, 2-3i ...
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1answer
31 views

ordered partition, block matrix given by $r_j \times r_j$ nilpotent Jordan blocks is nilpotent, rational canonical form, jordan canonical form

Let $F$ be a field. For an integer $n \ge 1$, and ordered partition of $n$ is a sequence $\underline{r} = \{r_1, \dots, r_m\}$ of positive integers such that $r_1 \le \dots \le r_m$ and $\sum r_j = ...
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2answers
54 views

Some questions about the Eigenvalues of this $4\times 4$ matrix

\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \end{bmatrix} The rank is $1$ as there is only $1$ linearly ...
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2answers
41 views

$A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive.

Let $A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive. Then which of the following statements is always false? A. ...
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0answers
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Null space, column space and row space be a line [duplicate]

For a 4x3 matrix can the nullspace, the column space and row space all be a line through the origin? For a 2x4 matrix can the nullspace, the column space and row space all be a plane through the ...
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2answers
17 views

Computing inner products with linearity in the first argument.

Consider the vector space $..$ $(\mathbb{P},\langle \cdot,\cdot \rangle)$ where the inner product is given by: $$$$ $\langle p(x),q(x) \rangle = \displaystyle\int_{-\infty}^{\infty} p(x)q(x)e^{-x^2} ...
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2answers
21 views

Question in regards to definition: finite dimensional

Do we denote a vector space as finite dimensional IF it has a basis, or do we say that it is finite dimensional if it's associated through an isomorphic transformation with a "number space", ie. ...
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38 views

Proof of this theorem: $ tr(A^-)=\sum_{i=1}^r \lambda_i^{-1} $

If $A$ is an $n\times n$ symmetric matrix with $r$ nonzero characteristic roots $ \lambda_1,\lambda_2,...,\lambda_r$ then $$ {\rm tr}(A^-)=\sum_{i=1}^r \lambda_i^{-1}. $$ Note: $A^-$ is generalized ...
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Prove or disprove the following statements. In linear systems, if the response of u (t) is y (t), then a) The response to Re [u (t)] is Re [y (t)]. [on hold]

Prove or disprove the following statements. In linear systems, if the response of u (t) is y (t), then a) The response to Re [u (t)] is Re [y (t)]. b) The response to Im [u (t)] is Im [y (t)].
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Thereom on Bases of Finite Vector Spaces

Please produce a proof of the following... Theorem: If $B_1$ and $B_2$ are two bases of a finite vector space $V$, then for all $\vec x\in B_1$, there exists $\vec y\in B_2$ such that $(B_1 - \{\vec ...
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Let u, v, w be three points in $\mathbb{R}^3$ not lying in any plane containing the origin.Then.. [on hold]

Let $u$, $v$, $w$ be three points in $\mathbb{R}^3$ not lying in any plane containing the origin. Then 1. $a_ 1 u +a_ 2 v +a_ 3 w = 0 \Rightarrow a_ 1 = a_ 2 =a_ 3 = 0$. 2 $u$, $v$, $w$ are mutually ...
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1answer
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Trace distance between “weighted” Hermitian matrices

The trace norm for a matrix $\mathbf{A}$ (also known as Shatten 1-norm) is defined as follows: $\|\mathbf{A}\|_1=\operatorname{trace}[\sqrt{\mathbf{A}^*\mathbf{A}}]$. It yields a useful distance ...
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1answer
26 views

what is the convex hull of such a matrix cone?

A matrix cone is in the following form: $M: = \begin{pmatrix} 1 \\ x\end{pmatrix}\begin{pmatrix}1 & x^T\end{pmatrix}$ where $x\in F$ , let $F = \{x: x\in [l,u]^n\}$ How to express the convex ...
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4answers
662 views

A faster way of calculating this determinant?

I'm doing a problem involving Cramer's rule, and one of the determinants I have to work with is as follows: \begin{vmatrix} 1&1&1\\ a&b&c\\ a^3&b^3&c^3 \end{vmatrix} So I ...
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2answers
28 views

How to show a set spans a space?

I've just started working with abstract algebra, and while the theory makes some sense, I have a bit of trouble figuring out the actual methods to complement the theory. For example, a base for a ...
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1answer
31 views

Order of $\mathrm{GL}_n(\mathbb F_p)$ for $p$ prime [duplicate]

While proving some facts about matrix group operations on finite fields, I stumbled across the following question: What is the order of the group of invertible $n\times n$ matrices over a ...
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1answer
39 views

trace inequality of positive definite matrices.

Assume $A,B \in M_n(\Bbb{R})$ are positive definite matrices, show that $$\text{Tr}(AB)\leq \text{Tr}(A)\text{Tr}(B) $$ I only prove it for $n=2$, it is straightforward calculate.but when $n \geq ...
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29 views

Maps from 3-D to 2-D

What is the name for a function which maps from a Euclidian 3D space to a 2D image? Basically imagine you have some 3D modeling software. I am talking about the function that maps from the 3D model, ...
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16 views

How to show if $\alpha$ is in the range than so is $T(\alpha)$ [on hold]

If I define $E_k \in L(V,V)$ by $E_1(\alpha)=-(x-2)^3$ $E_2(\alpha)=(x-1)(1-(x-2)-3(x-2)^2)$ and I let $W_k$ be the range of $E_k$. How would I show that if $\alpha \in W_k$ then $T(\alpha) \in ...
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1answer
24 views

Polynomial functions/basis

If I suppose $R \subset F$ and have polynomial functions $p_{k,j} : F \to F$ by $p_{1,0}(x)=(x-2)^3$ $p_{2,0}(x)=(x-1)$ $p_{2,1}(x)=(x-1)(x-2)$ $p_{2,2}(x)=(x-1)(x-2)^2$ and the polynomial ...
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I have an open note linear algebra exam soon. What notes should I print off the internet? [on hold]

I'm about to have my linear algebra final exam in 2 days. My professor is allowing us to use anything I can put on paper. These are the concepts I have to know: Linear Equations in Linear Algebra ...
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1answer
75 views

$T:V\to W$, both has same basis

Suppose $W,V$ have the same basis $\{u_1,u_2\}$ and that $T:V \to W$ is a linear transformation. Give an example (not the identity transformation) of a) $T = T^{-1}$ b) $T= T^2$ for a) $T=T^{-1} ...
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Is there necessary and sufficient condition on whether a symmetric circulant matrix is non-singular?

Is there necessary and sufficient condition on whether a symmetric circulant matrix is non-singular? I found many example supporting that positive symmetric circulant matrices which has at least two ...
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1answer
42 views

Is $x^3$ in the null space of the transformation $p(x) \mapsto xp(x)$?

Let $h: P_3 \to P_4$ be given by $p(x) \mapsto xp(x)$. Is $x^3$ in the null space ? Or is it in the range space ? Also, I am having difficulty finding the null space and the range of this map, can ...
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2answers
38 views

Show that V=W1⊕W2

Let $L_1$, $L_2$ be linear operators where $$L_1=L_1^2, \quad L_2=L_2^2 \tag{a}$$ $$L_1L_2=0, \quad L_2L_1=0 \tag{b}$$ $$I=L1+L2\tag{c}$$ Show that $V=W_1 \oplus W_2$. ($W_1$ should be the range of ...
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Find an isometric matrix such that

Find an isometric matrix $Q$ such that $$Q*\begin{pmatrix} -3 \\ 4\\ -5 \\ \end{pmatrix} = \begin{pmatrix} 2 \\ -4 \\ 6 \\ \end{pmatrix}$$
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Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
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1answer
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If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent

Let $T: V \to V$ with a minimal polynomial $z^n$, prove there's a vector $v$ such that $v, Tv, T^2v, ..., T^{n-1}v$ are linearly independent? The way I did it orginally was not allowed. No Jordan ...
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31 views

Adjoint linear operators and inner products question; why does $\langle T(x),T(x)\rangle =\langle T^*T(x),x\rangle $?

I have seen this multiple times in my textbook; $\langle T(x),T(x)\rangle=\langle T^*T(x),x\rangle$; why is this true? I know the definition of adjoint is if $\langle x,T(y)\rangle=\langle ...
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0answers
30 views

sequence of linearly independent vectors

Lets say that we have I linerly independent vectors $\{v_1,v_2,...,v_I\}$. And lets say that we have a sequence of vectors $\{x^k\}^k$, where $x^k=\Sigma_Ic_i^kv_i$. Lets say that the sequence of ...
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0answers
46 views

Prove that this defines an inner product on polynomial space [on hold]

An inner product on the polynomial space $P_2$ is a function that associates a real number, as denoted $\langle p, q \rangle$, to each pair of polynomials $p,q$ in $P_2$ and satisfies the following ...