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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Computing the exponential of the operator $ H(\textbf{x}) = \alpha \textbf{n} \times\textbf{x}$

Define the vector operator: $$ H(\textbf{x}) \equiv \alpha \textbf{n} \times\textbf{x}$$ For unit vector $\textbf{n}$ and some constant $\alpha$. We define further the operator: $$G \equiv I + H + ...
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68 views

Existence of a Surjective Linear Map $\mathbb{R}^3$ to $\mathbb{R}^4$?

Is it possible for there to be a surjective linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^4$? I feel like this isn't possible because $\mathbb{R}^3$ has fewer vectors than $\mathbb{R}^4$. ...
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39 views

How to find the complex roots of $x^2-2ax+a^2+b^2$?

How to find the complex roots of $x^2-2ax+a^2+b^2$? I tried using the quadratic formula: $$ x_{1,2} = \frac{2a \pm \sqrt {4a^2-4b^2}}{2} = {a \pm \sqrt {a^2-b^2}} = a\pm \sqrt{a-b}\sqrt{a+b}$$ ...
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228 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
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274 views

Derivative of a quadratic form

There is a Hermitian matrix $X$ and a complex vector $a$. I know that $a^HXa$ is a real scalar but derivative of $a^HXa$ with respect to $a$ is complex, $$\frac{\partial a^HXa}{\partial a}=Xa^*$$ Why ...
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259 views

Linear dependency of nilpotent matrices

I would like to prove that four $2\times 2$ nilpotent matrices are always linearly dependent, using the Cayley-Hamilton theorem or the minimal polynomial in some way. I think I have proved the ...
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294 views

Prove that $T,S$ are simultaneously diagonalizable iff $TS=ST$. [duplicate]

Definition: We say that $S,T$ are simultaneously diagonalizable if there's a basis, $B$ which composed by eigen-vectores of both $T$ and $S$ Show that $S,T$ are simultaneously diagonalizable iff ...
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448 views

Find a point on the line: $y=2x-5$ that is the closest to $P(1,2)$

This is our line: $f(x)=2x-5$ I have to find a point on this line that is the closest to the point $P(1,2)$. How do I go on about solving this? Should I use derivative and distance from the point to ...
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1answer
181 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
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108 views

Linear independence of matrices $I, A, A^2$

I want to prove that $I,A,A^2\:$matrices $\in M_{2\times 2}$ are $\textit {linearly independent}$. I consider the following matrices and their "corresponding" vectors: $I=\begin{pmatrix} 1 & 0 ...
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848 views

Proving determinants using properties of determinants

$$\begin{vmatrix} 1 & a^2+bc & a^3\\ 1 & b^2+ca & b^3\\ 1 & c^2+ab & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a^2+b^2+c^2)$$ we have to solve this by using the properties of ...
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148 views

Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...
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102 views

If $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$

I have looked here, but still I cannot understand how to get to equality. Let assume that the matrices are squared $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ is easy to show, but how can I ...
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3answers
250 views

How to deal with linear recurrence that it's characteristic polynomial has multiple roots?

example , $$ a_n=6a_{n-1}-9a_{n-2},a_0=0,a_1=1 $$ what is the $a_n$? In fact, I want to know there are any way to deal with this situation.
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269 views

Matrix Equation- solution

Sir, We have given $A= \begin{bmatrix}q_1 & q_2&q_3 \\ q_4 & q_5&q_6\\ q_7 & q_8&q_9 \end{bmatrix} \tag 1$. A is a matrix with determinant 1,orthogonal , invertible and ...
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149 views

How do you quickly find the eigenvalues of this matrix?

I have a final exam tomorrow, am sure a 3x3 eigen value problem like the one below is there. But I find it very hard to find eigen values without zeros in the matrix Show me how you do it quickly so ...
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158 views

I see some contradiction in the definition of orthogonal vectors

Let's look at the well-known definition of orthogonal vectors: Let $V$ be a vector space. Two vectors $x, y \in V$ are orthogonal to each other when the following condition is fulfilled: $$\langle ...
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1answer
154 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
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1k views

How to find exponential of triangular matrix

I'm studying for an exam and I can't find this in my notes or in the book, but it's on a past exam... Given $A = \begin{bmatrix}-1 & 1\\0 & -1\end{bmatrix}$, $e^{tA} = \begin{bmatrix}e^{-t} ...
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456 views

Show that the image of a linear transformation is equal to the kernel

Let $\phi$ be a linear transformation such that $\phi: V\to V$ We are given the following facts: $\dim(V) = 8$ $\dim(\mathrm{Im}(\phi)) = 4$ $\phi\circ\phi=0$ Show that $\mathrm{Im}(\phi) = \ker ...
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178 views

Why is tensor product of linear maps defined as $(S\otimes T)(v\otimes w)=S(v)\otimes T(w)$?

In my understanding, the definition of tensor product of linear maps cannot be directly derived from the definition of tensor product of vector spaces (or modules), since it's not clear what is the ...
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111 views

Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
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144 views

About inner product

I'm stuck in the next problem, Let $n$ a positive integer and $V=M_n(\mathbb{C})$. We define the inner product by $\langle A,B\rangle=\operatorname{tr}(A^t\overline{B})$. Let $W$ the subspace of $V$ ...
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1k views

Prove that the characteristic polynomial of a nilpotent matrix is $x^n$

How can I prove that the char.pol. of a nilpotent matrix is of the form $x^k$? I'm trying to do it by contradiction but assuming that $p_{xA}=a_0+a_1x+\dots+a_mx^m+\dots+a_nx^n$ seems not giving any ...
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71 views

$\det A \neq 0$. Prove that $\det A^* \neq 0$.

$A$ is matrix representing operator $\mathcal{A}$. $*$ is such operator that respects following equality: $(\mathcal{A}x,y)=(x, \mathcal{A}^*y)$; (I don't know what term is used in English). ...
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63 views

I need help with a proof showing $\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 $

So, I am dealing with the 2-norm and the projection is defined as the standard orthogonal projection, so far I have $$\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 ...
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64 views

Is $a^T M b = b^T M a$?

I am trying to reproduce a proof and I'm stuck at one point where it looks like this : $a^T M b + b^T M a = 2 a^T M b$ Therefore the only explanation I can get is this, if true : is $a^T M b = ...
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2k views

How to prove a set of vectors does not span a space.

Ok, so I'm a bit curios as to how you can prove a set does not span a vector space. For example, let ${S}$ be the vector set \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0\\ ...
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202 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
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218 views

Multiplying Adjacent Matrices?

My teacher hasn't explained it too well, so i'm looking for an explanation: $$A = \begin{pmatrix} 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0 \end{pmatrix}$$ $$A^2 = ...
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67 views

Why if $aX+c=bX+d$ then $a=b$ and $c=d$?

There is theorem in linear algebra. I forgot it!! But I remember something from it. Can you please give me a reference? It is related to something like this. If I have two polynomials ...
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4answers
85 views

Are there no solutions for $\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$?

I'm trying to solve an equation system using Gauss-Jordan. $$\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$$ So, first, the augmented matrix: \begin{bmatrix} 2&4&5\\ 3&6&6\\ ...
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149 views

Why is the $O$ (zero) matrix important?

In reading my linear algebra book I found it quite interesting that they made the following comment: One important property of addition of real numbers is that the number $0$ is the additive ...
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118 views

Prove that linear operator is a basis.

Prove if T : R n → R n is a 1−1 linear operator, and {v1, v2, . . . , vn} is a basis of R n, then so is {T (v1), T (v2), . . . , T (vn)}. I am not sure how I can show that. Any help or tips?
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146 views

Is the group of all determinants of all invertible $n \times n$-matrices isomorphic to $\langle\mathbb{R}^*, \cdot\rangle$?

I am doing Linear Algebra and Abstract Algebra simultaneously, and in Linear Algebra class, going through determinants, I thought of something interesting (for a freshman just learning the subjects, ...
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6answers
50 views

If $k$ cannot equal $0$ and $A$ is as given below, what is $A$ inverse?

$$ A = \pmatrix{ 1 & 0 & 2k\\ 0 & 1 & k\\ 0 & 0 & k } $$ I don't know what method to use to solve this problem as I haven't encountered and variable before when solving for ...
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123 views

Calculate the series: $\sum^\infty_{n=1}\frac{(-1)^n}{n^2}$ using dirichlet's theorem

This question was in my exam: Calculate the series: $$\sum^\infty_{n=1}\frac{(-1)^n}{n^2}$$. I answered wrong and the teacher noted: ...
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120 views

Question regarding sub-spaces of finitely generated vector space

Let $L_1,L_2$ be sub-spaces of finitely generated vector space. Prove that if $\dim(L_1+L_2)=1+\dim(L_1 \cap L_2)$, then $L_1 \subseteq L_2$ or $L_2 \subseteq L_1$. Unfortunately, I don't ...
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992 views

The kernel and image of $T^n$

I need help with this question: Let $V$ be a finite vector space where $ \dim V = n $, over the complex numbers and let $ T: V\to V $ be a linear transformation. Prove that $ V = \ker(T^n) \oplus ...
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658 views

Dimension of sum and intersection of vector space.

I am trying to understand the proof of the following: Suppose $U,W$ are vector subspace of $V$, then $\dim (U+W)+\dim (U \cap W)= \dim (U) +\dim (W).$ The proof goes like this: Let $S: V \rightarrow ...
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4answers
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Find the values of $x$ which makes $\det (A)=0$ without expending determinant

Find the values of $x$ which makes $\det(A)=0$ without expending determinant: Let $A$ : $$\begin{bmatrix}1 & -1 & x \\2 & 1 & x^2\\ 4 & -1 & x^3 \end{bmatrix} $$ How can I ...
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4answers
92 views

How to solve for matrix $A$ in $AB = I$

Given $B$ = $\begin{bmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1 \end{bmatrix}$ I know that $B$ is equal to inverse of $A$, how can I go backwards to solve for $A$ in $AB = ...
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4answers
116 views

Finding the characteristic polynomial of this specific $3\times3$ matrix

How can I find the characteristic polynomial of the following matrix: \begin{pmatrix} 0&-2&2\\-2&1&0\\2&0&-1 \end{pmatrix} please I need the details.
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1answer
565 views

What does this linear algebra notation mean?

I'm trying to prove that a particular $V$ is a $\Bbb{Q}$-vector space. The question says to take the element $0_V = 1$, the function $+_V : V \times V \to V$ given by the function $[x +_V y = xy]$, ...
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163 views

Eigenvalues of the Google Matrix

Let $c \in \mathbb{C}$ and $x, v \in \mathbb{C}^n$ satisfy $v^*x= 1$ (where the start means conjugate transpose). Given that a square ($n\times n$) matrix $A$ with complex entries has eigenvalues ...
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175 views

Linear Dependence Of A Sum

I had to prove the following notation: if $u1,u2,u3$ are linear depended then $u1+u2,u2+u3,u3+u1$ are also linear depended. I tried to contradict saying that I assume that P is right but Q is wrong ...
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1k views

Given two similar matrices $A$, $B$, is there a way to find an invertible matrix $P$ such that $A=P^{-1}BP$?

I was wondering if given two similar square matrices $A$ and $B$ would always be possible to find an matrix $P\in GL(n)$ such that $A=P^{-1}BP$. thank you!
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111 views

Complex numbers problem

I have to solve where n is equal to n=80996.
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1answer
418 views

Question about diagonal entries of inverse matrices?

Assume $A$ is a symmetric positive semidefinite matrix with diagonal zero and all other entries are less than one. Also assume $D$ is a diagonal matrix with all entries in diagonal are positive and ...
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2answers
573 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: $$ = ...