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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Given matrix $A$ such that $A^8+A^2=I$ prove that $A$ is diagonalized

Given matrix $A\in M_{n}(\mathbb{C})$ such that $A^8+A^2=I$ prove that $A$ is diagonalized. 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 and ...
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2answers
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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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I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $T(-2 x^2)= 3 x^2 + 3 x,\\T(0.5 x + 4)= -2 x^2 - 2 x - 3,\\ T(2 x^2 - 1) = -3 x + 2 .$ With respect to these three input vectors, ...
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3answers
48 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.) ...
1
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1answer
12 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
53 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
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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
24 views

I have to show $a_{nn} \neq 0$.

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} & ...
0
votes
1answer
14 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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29 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
72 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
60 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 ...
4
votes
2answers
42 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
votes
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
12 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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15 views

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
14 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
35 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
37 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
32 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
33 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
54 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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3answers
72 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
votes
1answer
24 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
24 views

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

I need to find the projection of $\v b = (1,1,1,1)$ 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
25 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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20 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 ...
0
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1answer
15 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 ...
4
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1answer
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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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0answers
21 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
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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
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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 = ...
2
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0answers
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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 = ...
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2answers
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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
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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 ...
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2answers
23 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 ...
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1answer
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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 ...
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Tensor independence

Let $(e_{i})$ be a basis in $V$, $( \epsilon_{i} )$ - basis in $V^{*}$ so that $\epsilon_{i} (e_{j})= \delta_{i}^{j}$ (Kronecker delta, $\epsilon_{i} (e_{j}) = 1 \Leftrightarrow i=j$, otherwise it's ...
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Why do the vectors perpendicular to [1, 1, 1] and [1, 2, 3] fall on a line, as opposed to a plane?

And what's the intuition here? This is question 6(c) in pset 1.2, Strang's Linear Algebgra, 4th Ed.
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1answer
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Let $T:U\rightarrow V$ be a linear map and suppose that $rank(T)=dim(U)=dim(V)=n$. Show that the are bases where the matrix is $I_n$

I found this problem that I cannot solve, but I believe is quite interesting. We have to state whether the statement is true or false. Let $T:U\rightarrow V$ be a linear map and suppose that ...
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2answers
28 views

Linear operator of infinite dimension

Let $T: V\rightarrow V$ a linear operator with finite dimension. If exists a linear operator $U: V\rightarrow V$ such that $TU=I$, prove that $T$ is invertible. Prove that if the ...
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84 views

Comprehensive Linear Algebra Text

Occasionally I come across a fact from linear algebra that I have not seen before. These facts are often obscured in search engines by either introductory texts or unrelated papers, and it is ...
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26 views

Prove statement about projection (linear map)

I am working on the following problem and do am not sure how best to approach it. Let $U$ be a vector space over a field $F$ and $p, q: U \to U$ be linear maps. Assume $p + q = \text{id}_U$ and $pq = ...
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0answers
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Generalized “softmax”

I'm looking for a function $f$ from $\mathbb{R}^n$ to $[0,1]^n$, similar to softmax in the sense that is satisfies these properties: $\sum_i f(x)_i = c$, where $c$ is a chosen constant (i.e., c=1 ...
4
votes
1answer
37 views

What operations can I do to simplify calculations of determinant?

My question is simple. Given an $n \times n$ matrix $A$, what operations can we do to the rows and columns of $A$ to make the calculation of its determinant easier? I know we can put it into row ...
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3answers
36 views

Understanding the difference between Span and Basis

I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for. I understand that the Span ...
0
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1answer
24 views

$A$ is positive definite if and only if $Q$ is invertible for every choice of $Q$

Note that if $A \in M_{n \times n}$, $A^{\prime}$ denotes the transpose of $A$. I proved the following theorem already: $A$ is nonnegative definite if and only if there exists a square matrix ...
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2answers
35 views

Determine which values of $\lambda \in \mathbb{R}$ cause the following vectors to be a basis

I am working on the following problem. Suppose that $\{v_1, v_2\}$ is a basis of a real vector space $V$. For which values of $\lambda \in \mathbb{R}$ is $\{w_+, w_\lambda\}$ a basis of $V$, where ...
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1answer
40 views

Solve linear algebra system [on hold]

Solve the linear equations $a·x = c$ and $a×x+b = 0$ for $x$ (which you may take to have components $x_1, x_2$ and $x_3$) if a $6= 0$ and $b$ are constant vectors and $c$ is a constant scalar. How ...