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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Appreciate help with solving a probability density function for its constant term

I am using StackOverflow a lot for asking and answering programming related questions, and I hope it is appropriate if I'd ask my question below on here on this sister-site. If not, please let me know ...
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Quaternion representation of rotations

How would one show that 1/3 turns correspond to the eight antipodal pairs among the 16 quaternions : $$ \pm \frac{1}{2} \pm \frac{i}{2} \pm \frac{j}{2} \pm \frac{k}{2} $$ knowing that the rotation ...
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Row Reduce Augmented Matrix

I am having issues actually row reducing it. What I initially get for the augmented matrix is: 0 1 -2 1 | 2 2 -2 4 -1 | 10 1 -1 1 0 | 2 1 0 1 0 | 9 But I am unsure how to actually use the ...
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217 views

Help with linear algebra network flow (picture)

I've been stuck on this problem for hours. I keep starting and stopping because I'm not exactly sure what I'm doing. The examples the teacher worked in class were much more straight forward. If ...
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Linear Algebra Standard Basis of $\mathbb{R}^3$

I have no idea what to do in this question, other then the fact that the formula $T$ is suppose to have some kind of $x$, $y$, $z$.
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For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
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What are the basis that span $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$

I have two questions: 1. What are the basis that span $\mathbb{R}^{2}$ Is it just $(0,1)$ and $(1,0)$? I read somewhere that it is $(0,1), (1,0), (1,1)$. But the 3rd one can be written as a linear ...
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Vectors and $\mathbb{R}^n$

For vectors $a$ $b$ and $c$ belong to $\mathbb{R}^n$, if $a·b=a·c$, then $b=c$?
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42 views

Models in Applied Mathematics 2nd order Systems

I have tried searching online but I don't seem to hit the right keyword to get an answer. Here is a 2nd order Probelm about Plants, the problem is that the plants Germinate in spring, bloom in ...
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303 views

Linear Transformation induced by the following matrix A

Suppose $T:\mathbb R^4\rightarrow\mathbb R^4$ is the transformation induced by the following matrix $A$. Determine whether $T$ is one-to-one and/or onto. If it is not one-to-one, show this by ...
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Real linear tranformation

When do we say that a transformation $T$ which takes the complex number field onto itself is real-linear? I need to know it for my homework but I can't seem to find the definition anywhere.
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diagonalisation and basis?

What are the conditions for something to be diagonalisable with regard to basis? I am trying to do this question: Let $V$ be a real $n$-dimensional vector space, and $T:V \rightarrow V$ be a ...
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104 views

Show a linear transform is self adjoint - check my answer

We are given $T:V \to V$ a normal linear transform (meaning $TT^*=T^*T$) We are also given $T^2=T$. Show that $T$ is self adjoint (meaning $T^*=T$). What I did I think I may have done something ...
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$T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$ [duplicate]

$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^*T=TT^*$. In addition $T^2=T$. Prove $T$ is self adjoint, that ...
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Find the projection matrix $P$

This problem projects $b=(b_{1},...,b_{m})$ onto the line through $a=(1,...1)$.The horizontal line $\hat{b}=3$ is closest to $b=(1,2,6)$ Find the projection matrix $P$. So $p$ $=(3,3,3) = P(1,2,6)$, ...
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Alternative coordinates for the complex plane $\mathrm{Re}[e^{-is}z]=a$, $\mathrm{Re}[e^{-it}z]=b $

I am defining coordintes on $\mathbb{C}$ using a "generalized" real and imaginary part. Here $a,b \in \mathbb{R}$. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ...
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Cross Product - Moments :: Dynamics 2

This problem is related to Cross Product - Moments :: Dynamics Please look at that link for the background on the problem I am faced with right now, I have linked a pdf of the book that I am using ...
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Find the projection of $b$ onto the column space of $A$

$A= \left[ {\begin{array}{ccccc} 1 & 1 \\ 1 & -1 \\ -2 & 4 \end{array} } \right] $ and $b = \left[ {\begin{array}{cccc} 1 \\ 2 \\ 7 \end{array} } \right] $ I ...
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350 views

Spectrum of a compact operator

If the spectrum of a compact operator is finite, I don't understand why $0$ has to be a member. I have proved that for all $\epsilon > 0$, there is only a finite number of eigenvectors which have ...
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100 views

Characteristic polynomial with coefficients c0=, c1=cn=1. Prove: $V = Ker(T) \oplus T(V) $

Question from final exam: $V$ is a vector space , $\dim V = n$, and $T:V\rightarrow V$ is a linear transformation. We assume that the characteristic polynomial of the linear transformation $$p_T(x) ...
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Is it true (or not) that the familly $\exp(rx)$ geenates the space of continuous functions from $\mathbb R$ to $\mathbb R$?

The family $\{ e^{rx} : r \in \mathbb{R} \}$ is a linearly independent set in the space of function from $\mathbb R$ to $\mathbb R$ (I know how to prove that). But I am wondering if it could generate ...
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Commutativity of two endomorphims

Let $f$ and $g$ be two endomorphisms of a real vector space $E$. I want to show that if $\ker(g)$ is stabilized by $f$ then $f\circ g=g\circ f$. Thank you for any hint.
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Existence of dual basis for an algebra with a nondegenerate bilinear pairing

Let $A$ be an algebra over a field $k$. Suppose that we have a non degenerate bilinear pairing $\beta:A \otimes A \to k$. Let $\{a_i\}$ be a basis of $A$. I would like to show that there exist a ...
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Is there a polynomial $f\in \mathbb Q[x]$ such that $f(x)^2=g(x)^2(x^2+1)$

I was asked the following question: $g\in \mathbb Q[x]$ is a polynomial (not the zero polynomial). Find $f \in \mathbb Q[x]$ such that $f(x)^2=g(x)^2(x^2+1)$ or show that such an $f$ does not exist. ...
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Simple yet confusing: if $ f^2(x)=g^2(x)(x^2+1) $ then $gcd( f^2(x),g^2(x))=(x^2+1)$?

As mentioned in the title: f(x) and g(x) are polynomials above the Rationals field. if $ f^2(x)=g^2(x)(x^2+1) $ then does it mean that $ gcd( f^2(x),g^2(x))=(x^2+1) $? or maybe it isn't the ...
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if $(a+\sqrt{a^2+1})$ and $(b+\sqrt{b^2+1})$ are converse then prove that a and b are opposites

$(a+\sqrt{a^2+1})\,(b+\sqrt{b^2+1})=1$ is supposed to equal: $b = -a$ but how do i get that? I've been trying to solve for like 2 days now.
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transformation and eigenvalues/eigenvectors

Define $S: M_n(R) \to M_n(R)$ by $S(A) = A^T.$ Prove that $S$ has only two distinct eigenvalues and that its eigenvectors span $M_n(R).$ I've noticed this: $$S(A) = A^T.$$ If I apply $S$ again, I ...
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Matrix diagonalization

Does any one have any idea how to diagonalize the following matrix: \begin{equation} \begin{pmatrix} 0 & a & 0 \\ a & 0 & b \\ 0 & b & c \end{pmatrix} \end{equation} The ...
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1answer
37 views

Nowhere dense in $\mathbb{C}^n$

$S=\{ A^k X \mid \ k \in \mathbb{N} \cup \{0\} \ \}$ where $A \in M_n (\mathbb{C}) $ and $X \in \mathbb{C}^n$. Show that $S$ is nowhere dense in $\mathbb{C}^n$.
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Question on linear algebra mappings

If $T:R^m\to R^n$ is a linear transformation, show that there is a number $M$ such that $|T(h)|\leq M|h|$ for $h\in R^m$.
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380 views

What properties should a matrix have if all its eigenvalues are real?

Recently, I’m trying to prove all the eigenvalues of a class of matrices are real. The matrices are complex and not hermitian. The problem for me is I don't know any properties for a matrix with all ...
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Prove that the nxn all 1 matrix (apart from diagonals which are 0) has -1 as an eigonvector with multiplicity n-1

I have no clue how to get started with this. Induction or something? The general formula for the determinant doesn't appear too helpful. Thanks for any help
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439 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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1answer
44 views

Determinants, traces and isomporphism of graphs

Question Prove that if A,B are adjacency matrices of two graphs, and their traces or determinants are not equal then the graphs are not isomorphic. Thoughts I know that 2 graph are isomorphic iff ...
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80 views

Show that $0$ and $1$ are the only real eigenvalues of $A.$

Let $A\in M_n(\mathbb R)$ such that $A^2=A^T.$ Show that $0$ and $1$ are the only real eigenvalues of $A.$ All I can see is that $\det A=0$ or $1.$ I can't proceed any further.
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Why are linear maps/transformations usually written with bases for the matrix representation?

If $T\in L(V,W)$, and we want the matrix representation, why do we usually choose to write $T$ in terms of the basis vectors of $V$ and $W$? Thanks!
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396 views

Eigenvalues of 3x3 Covariance Matrix, Geometric Interpretation

Problem Definition I would like to code an algorithm for decomposing a covariance matrix into its eigensolution (set of eigenvalues and corresponding eigenvectors. In my specific case I want to deal ...
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1answer
65 views

Computing the kernel of a homomorphism from a free $\mathbb{Z}$-module

Given a finitely generated free $\mathbb{Z}$-module $N$ and a homomorphism $\varphi:N\to M$, is there an easy way to compute the kernel of $\varphi$? Since the kernel is also a free ...
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1answer
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Given matrix A with eigenvalue $\lambda$ and corresponding eigenvector x, prove $A^k$ has eigenvalue $\lambda^k$

Given matrix A with eigenvalue $\lambda$ and corresponding eigenvector x, prove $A^k$ has eigenvalue $\lambda^k$ for the same eigenvector x for any positive integer k. Can I just use the eigenvalue ...
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Are positive semi-definite matrices always covariance matrices?

This may be trivial. While covariance matrices of random variables are positive semi-definite, does the converse hold true as well, that positive semi-definite matrices are also valid covariance ...
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Linear mapping between a non-orthogonal basis and an orthogonal basis?

Consider a set of $n$ linearly independent $d$-dimensional vectors $\left\{\vec{a}_i\right\}_{i=1}^{i=n}$ that span the vector space $V$ and that are not in general orthogonal with respect to the ...
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Prove this is a subspace of V

Let T: V $\to$ W be a linear map between vector spaces and let N be a subspace of W. Define $T(N) := {v∈V : Tv ∈ N}$. Prove that T(N) is a subspace of V. I know the properties that a subspace must ...
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Matrix of linear maps

I need a bit of clarification for an assignment question that I have. Let T: *F*$[t]_n$$\to$*F*$^2$ (where *F*$[t]_n$ represents polynomials of degree n) given by $T(f) = (f(1) , f(2))$. I am asked ...
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The real Matrix of change of basis. Not really. Only in $\mathbb{R}^n$.

Suppose we have an $\mathbb{R}$-vector space $E$, $\text{dim}(E)=n$, and two bases $\alpha:=\{v_i\}$ and $\beta:=\{w_i\}$ of it. We can consider the maps to $\mathbb{R}^n$ given by the coordinates ...
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1answer
86 views

Jacobi polynomials

We define the inner product on the space $\Bbb R[x]$ by $$\langle P,Q\rangle=\int_{-1}^1P(x)Q(x)(1-x^2)^\alpha dx$$ where $\alpha>-1$. I need to prove that for all $n\in\mathbb N$ ...
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Linear algebra and special relativity

I'm going over an exam I had a couple months back, over the exercises I didn't manage to get right and I'm kinda stuck with the following subtask: Let $\xi$ be a 4-vector with the Minkowski scalar ...
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194 views

How to construct the subring generated by a set, T?

I'm trying to find a constructive way of describing the subring generated by some subset, T, of a ring R. I think I could describe it as all finite sums of finite products of elements of T, but I ...
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78 views

Prove: A complex matrix with a rank of 1 is diagonizable iff its' trace is not $0$.

So far: The fact that the rank is 1 tells me that there are $n-1$ eigenvectors that are linearly independent with an eigenvalue of $0$. If the matrix is diagonizable, it's similar to a diagonal ...
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87 views

How to compute the diagonal matrix for this problem?

I did find the basis but I have no clue in solving the diagonal matrix part of the problem. Could someone please help me?
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589 views

Determinant of a finite-dimensional matrix in terms of trace

I have noticed that for the case of 1x1, 2x2 and 3x3 matrices $A$, $B$, I can write the determinant of their commutator $C=[A,B]$ in terms of traces: 1x1 matrices $A$, $B$: $$\det(C)=\text{tr}(C)$$ ...