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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Can a matrix be similar to more than one matrix?

I have a little query about similar matrices I've been struggling with. Suppose I have a 5x5 diagonal matrix A with 5 distinct eigenvalues as entries in the main diagonal. The question is, to how ...
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29 views

Reflection matrix and algebraic multiplicity

Let $Q\in\mathbb{M}_4(\mathbb{R})$ a reflection matrix onto $R(A)$ subspace, where $A\in\mathbb{M}_{4\times 3}(\mathbb{R})$ is defined by ...
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4answers
33 views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
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1answer
26 views

Show that the subset $S$ in $\mathbb{R}_3$ is a subspace.

Show that the subset $S$ in $\mathbb{R}_3$ defined by $S=\{(a,b,c) \in \mathbb{R}_3 \text{ such that } a+b=c \}$ is a subspace. I'm having trouble adapting the definition of subspace with the part ...
3
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1answer
38 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
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1answer
14 views

Is it possible to find a vector that is orthogonal to this set?

I have a set of four vectors in $\mathbb{R}^4$: $\{ \vec v_1, \vec v_2, \vec v_3, \vec v_4 \}$ The first three are linearly independent, but $ \vec v_4 $ is a linear combination of the others. Is it ...
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1answer
26 views

demonstrate that $v_3 \perp (v_1-v_2)$

$\bar{v}_1 \perp (\bar{v}_2-\bar{v}_3)$ and $\bar{v}_2 \perp (\bar{v}_3-\bar{v}_1)$ therefore $\bar{v}_3 \perp (\bar{v}_1-\bar{v}_2)$ By applying the dot product to $\bar{v}_1$ and ...
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1answer
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3answers
45 views

How are these expressions equivalent?

I saw that $${y^2\over y^2+1} = 1 - \frac1{y^2+1}$$ but I can't see how, wolfram alpha agrees but I'm still not seeing it.
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0answers
15 views

Polar decomposition varient

I have a factorisation to do, and I think that a varient of Polar decomposition will give me what I need, although I'm not sure of the exact form. I have \begin{equation*} \mathbf{y} = ...
2
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1answer
119 views

Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0

Given a matrix $A$ find a two dimensional subspace $V \subset\mathbb{R}^4$ for which $\forall x \in V : x^TAx=0$ $$A = \begin{pmatrix}1&2&0&1\\ 2&3&1&1\\ 0&1&0&1\\ ...
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2answers
41 views

Rotation matrix

I'm finding different results for the 3D rotation matrix in the XY plane from different sources and I was hoping for someone to help clarify. In my "applications of vector calculus" book, the matrix ...
7
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1answer
77 views

$A$ is diagonalizable if $A^8+A^2=I$

Given a matrix $A\in M_{n}(\mathbb{C})$ such that $A^8+A^2=I$, prove that $A$ is diagonalizable. So let $p(x)=x^8+x^2-1$ and we know that $p(A)=0$. The next step would be to show that the algebric ...
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2answers
22 views

methods of constructing a matrix from its null space span

I have a matrix of size $4\times3$ and its null-space span is $\{(1,2,3), (2,5,7)\}$. How can I find the original matrix? It is not obvious from the span which vectors are free.
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2answers
34 views

I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $$\begin{align} T\left(-2 x^2\right) &= 3 x^2 + 3 x \\ T(0.5 x + 4) &= -2 x^2 - 2 x - 3 \\ T\left(2 x^2 - 1\right) ...
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3answers
59 views

Self-Study Linear Algebra book for a complete understanding

I recently took an introductory class on linear algebra (covered solving linear systems, determinants, eigenvectors, diagonalization, some vector spaces, basis and combinations, transformations etc.) ...
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2answers
25 views

Algebric and geometric multiplicity and the way it affects the matrix

Given a matrix $A$. Suppose $A$ has $\lambda_1,\dots,\lambda_n$ eigenvalues each with $g_i$ geometric multiplicity and $r_1,\dots,r_n$ algebric multiplicity, $g_i\leq r_i$. Given this information ...
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3answers
67 views

Why does $\frac{1}{{\left\| {\left| {{A^{ - 1}}} \right|} \right\|}} \le \left\| {\left| B \right|} \right\|$?

Let $A,B \in {M_n}$ suppose that the following statements are true: $A$ is nonsingular, $A+B$ is singular, $\left\| {\left| . \right|} \right\|$ is matrix norm. Why is it true that: ...
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0answers
16 views

Find a generator for vectorial subspace

S = {$(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0$} $c+d-i(2ia)=0$ $c+d+2a=0$ $c=-d-2a$ $(a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1)$ Is this solution correct?
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1answer
27 views

I have to show $a_{nn} \neq 0$. [on hold]

Let $D$ be an algebraic division ring with center $F$. $A,B$ are upper triangular matrices in $M_n(D)$. let $ A=\begin{pmatrix} a_{11}&a_{12} & \ldots &a_{1n}\\ 0 & a_{22} & ...
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1answer
15 views

To find nullity of a surjective linear mapping

Let $T:U \to V$ be a surjective linear mapping and $dim(U)=6$,$dim(V)=3$.Then a) $dim(ker$ $T)$ is greater than $4 $ b) $dim(ker$ $T)$$ = 4 $ c) $dim(ker$ $T)$ is greater than $3 $ d) $dim(ker$ ...
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3answers
43 views

How to solve these simultaneous equations using numerical methods?

How to solve these simultaneous equations for $\alpha$ and $\lambda$ using numerical methods? $\lambda * [(\frac{3}{4})^\frac{-1}{\alpha} - 1] = 11$ $\lambda * [(\frac{1}{4})^\frac{-1}{\alpha} - 1] ...
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1answer
76 views

For which $\beta \in \mathbb{C}$ is the matrix $A=\bigl(\begin{smallmatrix} 0&1\\1&\beta \end{smallmatrix}\bigr)$ diagonalisable?

I have got a question refering to the following problem. Let $K=\mathbb{C}$. For which $\beta \in \mathbb{C}$ is this matrix diagonalisable? $$A=\pmatrix{0&1\\1&\beta}$$ I think that it is ...
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1answer
63 views

I cannot find the relation between $(I-A)x=v$ and $((I-A)^2)x=0$.

This question from my textbook: Solve $(I-A)x=v$ where $v$ is some $4\times 1$ matrix, $I$ is the identity matrix, and $A$ is some $4\times 4$ matrix and hence solve $((I-A)^2)x=0$. I cannot ...
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2answers
43 views

Let $T:V\rightarrow V$ be linear and injective. Prove that if $V$ is finite-dimensional then $T$ is surjective.

Could you help me solving this problem? Let $T:V\rightarrow V$ be linear and injective. Prove that if $V$ is finite-dimensional then $T$ is surjective. You are not allowed to use rank nullity ...
5
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1answer
39 views

What does a linear equation with more than 2 variables represent?

A linear equation with 2 variables, say $Ax+By+C = 0$, represents a line on a plane but what does a linear equation with 3 variables $Ax+By+Dz+c=0$ represent? A line in space, or something else? On ...
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0answers
17 views

Cross product of the gradient of two functions

I am having a bit of a confusion with some claims I keep finding on a book of Fluid Dynamics. Let's say we have two functions in 3-D space, $f(\mathbf{x})$ and $g(\mathbf{x})$, with the following ...
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MATLAB related query [on hold]

How we can execute a program in MATLAB of a perpendicular line passing through the mid point of a line segment.Plz help me
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0answers
18 views

Understanding Extension of Scalars in a Vector Space

$\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector ...
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4answers
39 views

Cauchy-Schwarz, how does it work in this example?

Consider $\|\cdot\|_2$ such that $\|x\|_2 = \left(\sum_{i=1}^n |x_i|^2\right)^{1/2}$. Let $A \in \mathbb{R}^{n\times n}, x\in \mathbb{R}^n$, then $$\begin{align} \|Ax\|_2^2 & = \sum_{i=1}^n ...
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2answers
40 views

find the rank of a linear mapping such that $T^2=0$

Let $T:\Bbb R^6\to\Bbb R^6$ be a linear mapping such that $T^2=0$.Then which one is true? a)Rank$(T)$ is less than or equal to 3 b)Rank$(T)$ is greater than 3 c)Rank$(T)$ is equal to 5 ...
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1answer
35 views

relation between trace of product and sum of matrices?

Given A and B positive definite matrices. Is there an inequality relation between trace(AB) and trace(A+B) ?
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1answer
35 views

Proof Norm is Continuous

Someone just asked me why the norm of a normed space is continuous, and the answer I gave them satisfied them, but I'm not sure if it should. Something seems amiss. Let $\rho: X \to \mathbb{R}^+_0$ ...
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1answer
57 views

Finding the characteristic polynomial of $A^2$ given the characteristic polynomial of $A$

To find the characteristic polynomial of the matrix $A^2$, would I just compute $$(\lambda^2+4\lambda-5)^2 ?$$
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5answers
164 views

Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$

We want to find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$ for an arbitrary "n". I have tried writing out a few elements of the sequence as $n \to ...
2
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1answer
26 views

Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
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0answers
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I have 4 in-equations with 4 variables in each of the in-equations. how to find the minimum value of each variable?? [on hold]

Please tell me the answer with solution. I don't know how to start it.completely blank.
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1answer
30 views

Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
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1answer
26 views

Find the linear transformation that is a reflection through the line $x=y$

Which of the following $2\times 2$ matrices corresponds to a linear transformation that is a reflection through the line $x=y$ in $ \Bbb R^2 $ ? a) $\begin{pmatrix} 1 & 0\\ 0 & -1 \\ ...
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0answers
22 views

Any program that turns vectors to orthogonal? [on hold]

Are there any sites that can transform S(set of vectors) into an orthogonal basis for R^n? I want to know if I did my problem correctly and would like verification. my vector set is [1 ,2, -1][1, 3 ...
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1answer
16 views

For a matrix $O$ containing columns which are an orthonormal basis for a column space, why does $O^{\prime}O = I$?

Theorem: let $\{o_i\}_{i \in \{1, 2, \dots, r\}}$ be an orthonormal basis for the column space of a matrix $X$ and let $O = \begin{bmatrix}o_1 & o_2 & \cdots & o_r\end{bmatrix}$. Then ...
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1answer
95 views

Suppose $AB=BA$ and $A^{1965}=B^{2015}=I$. Prove that $A+B+I $ is invertible.

Supppse $A $ abd $B $ are matrices, $AB=BA $ and $A^{1965}=B^{2015}=I $. Prove that $A+B+I $ is invertible. I want to prove that $(A+B+I)C=I $ I have no idea how to start. Can any one give some hint? ...
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2answers
39 views

QR decomposition proof

Let $A\in\mathbb{M}_{m\times n}(\mathbb{R})$ with $m>n$ and $rank(A)=n$ and take the decomposition $A=QR$ with $Q\in\mathbb{M}_{m\times n}(\mathbb{R})$ a orthogonal matrix and ...
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0answers
18 views

Discrete Fourier Time Question

Assume that $x[0]=1, x[1]=1, x[2]=1, x[3]=1, x[n]=0$ for $n \geq 4$, find the DFT of $$\{x[n]\}=( x[0], x[1], x[2], x[3] )$$. My method of doing this is to use the DFT formula as defined here: ...
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2answers
23 views

System of linear equations: get approximate solution with non-negative coefficients

I'm looking for a process or algorithm to help me with the following problem. I have the following vectors in $\mathbb R^{3}$: $$ \vec m_3 = \begin{bmatrix} 51.8\\ 2.9\\ 22.3 \end{bmatrix}, \vec a = ...
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0answers
20 views

An extension of Kato's Selection Theorem?

One formulation of the well-known Kato Selection Theorem states that, given an analytic family of $n \times n$ complex, symmetric matrices $M(t)$, one can choose an orthonormal basis $\{e_i(t)\}_{i = ...
2
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1answer
23 views

Composing Linear Transformations

Hello and thank you in advance; The problem: "Let V be a vector space and T a linear operator $T:V\rightarrow V $, show that $$[T^m]_B =[T]_B^m$$ Where $B$ is a basis(any) of $V$ and $T^m=T\circ T ...
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1answer
17 views

Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that every eigenvalue of $B^2$ is the square of some eigenvalue of $B$.

I would like to ask you for some help in the following problem: Suppose that a matrix $B$ is diagonalisable over $\mathbb{C}$. Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that ...
1
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2answers
24 views

direct sum of vectors

$$U = \{(x,y,z,t) \in \mathbb{R}^4 | x + 5y + 4z + t = 0 , y + 2z + t = 0 \} $$ $$W = \{(x,y,z,t) \in \mathbb{R}^4 | x + z + 3t = 0, 2x-3y-4z+3t = 0\} $$ $U \oplus W = \mathbb{R}^4$? This is my ...
0
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1answer
19 views

Isomorphism between vector spaces of linear transformations

Let $V,W$ vector spaces over the field $F$,and let $U: V\rightarrow W$ an isomorphism between them. Prove that the linear transformation $\mathcal{U}:\mathcal{L}(V,V)\rightarrow ...