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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Polynomial rings and quotients

Let $F$ be a field, $x$ an indeterminate in $F$, and $f(x)\in F[x]$ a polynomial with degree n. If the "Overline" denotes the canonical homomorphism from $F[x]\rightarrow F[x]/<f(x)>$, then, how ...
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Proving A ~ CB if A = BC and 0 is not an eigenvalue of B

I am trying to prove that, if A, B, and C are $n \times n$ matrices, $0$ is not an eigenvalue of $B$, and $A = BC$ that $A$ is similar to $CB$ I know that I have to get to $A = SCBS^{-1}$ for some $n ...
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$L\cdot M:=\pi_{r+s}(L\otimes M)$, then $((f_i\cdot f_j)\cdot f_k)\cdot f_l=?$

I'm reading Kenneth Hoffman's "Linear Algebra", Ed 2. In $\S5.7$ "the Grassman Ring" it tries to explain the way lead to the definition of exterior product (a.k.a. wedge product). However I got some ...
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Is there an algebraic solution to this problem?

The base of my pool cartridge filter tank is composed of an arc and a chord and has a total perimeter of 49.5 inches. The length of the chord portion of the shape is 7 inches. The arc portion has a ...
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Dissimilar Jordan matrices

So I need to find all dissimilar Jordan matrices J with the minimal polynomial: $x^3(x^2-1)^3$ and characteristic polynomial: $x^4(x^2-1)^4(x+1)^2$. So my question is, since the minimal polynomial ...
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Which beta distribution(s) has a variance `V` and a skew `S`?

Let X be a beta distributed random variable with parameters $\alpha$ and $\beta$, variance V and skew ...
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30 views

Are the operators $X$ and $Y$ equal to one another?

Suppose we've got two linear maps $\ X:\mathcal{H}\rightarrow\mathcal{H}\ $ and $\ Y:\mathcal{H}\rightarrow\mathcal{H}$, where $\mathcal{H}$ is some finite-dimensional Hilbert space. Let's say ...
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Invertible matrix polynomial without constant

How can I prove that if a polynomial Q without constant term such that $$ ||I-Q(A)|| <1 $$ then, A is invertible
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n x n matrix has the same eigenvalues as its transpose

I am trying to prove that a $n \times n$ matrix $A$ and $A^T$ have the same eigenvalues. I can prove that $A$ and $A^T$ have the same entries on the diagonal, but I am not sure where to go from ...
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41 views

Proving that eigenvalues are real and eigenvectors are orthogonal

If $A$ is a real symmetric matrix, can I prove that all of the eigenvalues of $A$ are real and that all eigenvectors associated with distinct eigenvalues are orthogonal? If so, where do I start to ...
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Showing there exists an $m\times n$ matrix $A$ with $\text{Row}(A) = U$ and $\text{Col}(A) = V$ where $U$ and $V$ are subspaces.

Let $F$ be a field. Let $U$ be a subspace of $F^n$ and let $V$ be a subspace of $F^m$. Suppose that $\dim U = \dim V$. Then there exists an $m\times n$ matrix $A$ with entries in $F$ such that ...
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Find all values of a and b for which the following system has non-trivial solutions.

I am having trouble with this Linear Algebra problem and I don't seem to find any similar problems on this site. Just a short description or a small hint is highly appreciated. Find all values of a ...
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36 views

Understanding operator norms

$A=\left(\begin{array}{cc} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{array}\right)$ Please help explain the fault in ...
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179 views

Change of basis - is the dot product method correct?

The traditional way of expressing a vector in a differnet basis relies on change of basis matrix (see here) I'm not sure why almost no book or online texts mention this, but there's another method. ...
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existence of a linear operator which extends linear map

Suppose $W$ is finite dimensional and $T_1,T_2 \in L(V,W)$. Prove that null $T_1 \subset$ null $T_2$ if and only if there exists $S \in L(W,W)$ such that $T_2=ST_1$ I proved that if there exists $S ...
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35 views

Proving $0$ is an eigenvalue of a matrix

I am trying to prove that an $n\times n$ matrix $A$, with $m$ being a positive integer such that $A^m = 0$ matrix has $0$ as an eigenvalue, but I'm not sure where to start. I also thought that $0$ was ...
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Why do we keep the LCM modulo in the Chinese Remainder Theorem?

I'm doing my homework and I'm struggling to get an answer. I'm taking number theory and we're working on a problem to solve congruences. We've got: $ x\equiv 1 \pmod{5}\\ x\equiv 3 \pmod{8}\\ x\equiv ...
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208 views

Maximize trace of matrix equation given two constraints

Let $\mathbf{Q}$ be a rotation matrix and $\mathbf{A}$ and $\mathbf{B}$ be two real-valued matrices of the same size. I want to maximize the function $$ f(\mathbf{Q})=tr\;\mathbf{QA} \qquad ...
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Finding determinants by inspection?

I'm supposed to "use properties of determinants to evaluate the determinant by inspection" on this matrix: $$\begin{bmatrix} 4 & 1& 3\\ -2 & 0 &-2 \\ 5 & 4 & ...
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111 views

İf x is diagonalizable then ad(x) is also diagonalizable

I start to study lie algebras from K. Erdmann, Mark J. Wildon-Introduction to Lie Algebras and i try to solve question below but actually i can't see .How can i start ? Give me hint please Let ...
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Demonstrating if a transformation is linear by using linear independence/dependence

Can examining the linear dependence/independence of the vectors in the domain and codomain demonstrate that a transformation is linear? I think that if there are linearly independent vectors in the ...
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37 views

Is there a linear vector field satisfying these properties?

I had a small question if this is possible while reading up a book. I have a linear vector field $v(x) = Ax$ on $\mathbb{R}^2$. Let $x_0$ be a point such that $v(x_0) \not= 0$. Is there a linear ...
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Orthogonality and Kernel

Here: http://ltcconline.net/greenl/courses/203/MatrixOnVectors/leastSquares.htm "Notice that $\mathbf{b} - \text{proj}_W\mathbf{b}$ is in the orthogonal complement of $W$ hence in the null space of ...
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From a set of vertices, find smallest polytope enclosing another point

Out of a set of vertices $V=\{\vec v_i\in \mathbb R^D\}$, I am constructing a piecewise linear interpolating function $f:\mathbf{conv}(V)\rightarrow R$ as follows: given a point $\vec d\in ...
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Problem with orthogonal mapping

We have an orthogonal mapping $A$ on an Euclidean space and two linearly independent vectors $x,y$ such that for some $a,b\in \mathbb R$, $b\neq 0$, $$ Ax=ax-by; $$ $$ Ay=bx+ay. $$ Since $x,y$ form ...
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19 views

How to write product of an array in vector-matrix form?

Suppose that, we have a vector $\boldsymbol{\beta}=[\beta_1,\ldots , \beta_n]$. Can we express $\prod_{i=1}^{n} \beta_i^2$ using vector/matrix operations on the vector $\boldsymbol{\beta}$? For ...
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71 views

Is $T(M)=PMP^{-1}$, where $P=\begin{bmatrix}2&3\\5&7\end{bmatrix}$ linear? If so, how to prove?

If I define $\vec{v}=\begin{bmatrix}a\\b\end{bmatrix}\text{and }\vec{w}=\begin{bmatrix}c\\d\end{bmatrix}$, I end up getting ...
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57 views

Prove statement about relation of trace and eigenvalues

I am studying linear algebra. I saw on a website the statement that if $\lambda^n+c_{n-1}\lambda^{n-1}+\cdots+c_1\lambda+c_0$ is the characteristic polynomial of a $n\times n$ matrix $A$ then ...
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Dimensions of spaces over different fields

We view $\Bbb C_2 = \{{w \choose z}:w,z\in \Bbb C\}$ as a vector space over $\Bbb C$, $\Bbb R$ and $\Bbb Q$. Let $x_1={i \choose 0}$, $x_2={\sqrt2 \choose \sqrt5}$, $x_3={0 \choose 1}$, $x_4={i\sqrt3 ...
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Prove $f$ is diagonalizable iff $V=W \oplus Z$ where $W,Z \subseteq V$ are $f$ invariant

Let $K$ be field algebraically closed, $f\in End(V)$ prove: $f$ is diagonalizable $\iff$ $\forall W \subset V$ invarant under $f$ exist $Z \subset V$ invariant under $f$ such that $V=W\oplus Z$ i ...
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All Eigenvalues of the operator $L(v)= L^2(v).$

Let $L: \Bbb R^n \rightarrow \Bbb R^n $ be a linear operator with the property $L(v)= L^2(v).$ Then find all the eigenvalues of the operator $L$. My attempt: $L(v)= L^2(v)=L^3(v)=L^4(v)$ and so on. ...
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Question about inequality in linear algebra

$V$ is inner product space. $u, v \in V$ are two orthogonal vectors. Prove that $\|v-u\| \geq \|v\|$. Because $\|v-u\|, \|v\| \geq 0$ it's enough to prove that $||v-u||^2 \geq \|v\|^2$. ...
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linear transformations same field

why should linear transformation involve vector spaces over same field? definition of a linear transformation: Let U and V be 2 vector spaces 'over the same field K'.So what happens if the vector ...
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Linear Independence and Bases

Hey everyone, I'm confused as to how I can find the basis from the following question. I know that basis has to be linearly independent and it has to span all of $R^4$. I know that each $x$ is it's ...
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Linear Dependency

Hey everyone, I'm having trouble with this problem. Asking for linear dependency means that the determinant has to be 0. So I'm assuming we can take the three vectors given and put them in a matrix? ...
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linear independence

Hey everyone, I'm having trouble with this problem. I set up two equations to be: $1+k = 0$ and $k + 2k+48$ and I got that k = -24 and or -1. But the answers don't work so my logic is clearly ...
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89 views

trace inequalities: linear algebra

If S is any $n \times n$ real, symmetric, invertible matrix and D is any $n \times n$ diagonal matrix such that $0\prec D \prec I$ then does there exist a constant $\gamma$ such that: ...
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Is $\sqrt{-x^2-\frac{1}{x}}$ a rational function?

I have to construct a rational function with the range being $[-1,0)$, which is pretty much just $-1$. I came up with the solution $\sqrt{-x^2-\frac{1}{x}}$. It works for the range, but I'm not sure ...
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How do I solve this system of equations using back substitution?

$$ \begin{cases} -x_5+x_4=3\\ -x_5+2x_4+x_3=4\\ 3.5x_4+5x_3+2x_3+2x_2+x_1=3.5 \end{cases} $$ I know how to back substitute to solve for systems where you go you 1 equation down and there is an one ...
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60 views

Commutativity of scalar/vector product: $a\mathbf{v}=\mathbf{v}a$ for all $a \in F$ and $\mathbf{v} \in V$

There are traditionally 8 axioms to check whether a set $V$ together with a field $F$ constitute a vector space. A common list of axioms can be found here. Missing from the list, however, is a ...
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How is this map injective?

Let $X$ be a (real or complex) vector space, let $X^{*}$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all linear functionals defined on ...
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Linear Algebra. Bonus question Final Exam

This is from a practice final exam. I was wondering about the the 2nd part of A). It states that when v can't = O, it will form a basis. I'm having a tough time understanding that. How is that ...
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On the determinant of a certain matrix over the polynomial ring of $n$ variables over a field

Let $A = k[x_1,\dots, x_n]$ be a polynomial ring over a field $k$. Let $\sigma_1,\dots,\sigma_n$ be distinct permutations of the set $\{1,\dots,n\}$. Is the determinant det$(x_{\sigma_i(j)})$ ...
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Finding a basis for subspace of polynomials

Let $V=\mathscr{P}_{3}$ be the vector space of polynomials of degree 3. Let W be the subspace of polynomials p(x) such that p(0)= 0 and p(1)= 0. Find a basis for W. Extend the basis to a basis of V. ...
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Find an orthogonal matrix such that its first line is $\frac{1}{5},\frac{2}{5}$

An orthogonal matrix is one matrix $A$ such that $A^t = A^{-1}$. So what I did: Suppose: $$A = \begin{bmatrix}\frac{1}{5}&\frac{2}{5}\\x&y\end{bmatrix}$$ Then: ...
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87 views

Linear algebra gram-schmidt

Will the reordering of the original basis before starting the Gram-Schmidt process lead to the same orthogonal basis? Is there an obvious proof for this one or is this clear already?
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Spectral radius of matrix from SOR method

Suppose we write a matrix $A = L + D + U$ with lower triangular, diagonal and upper triangular parts. When trying to solve the equation $Ax=b$, we use a successive overrelaxation technique such that ...
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64 views

Frobenius Norm to L2 norm Problem

Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that $$ ||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2. $$ I am having ...
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What is the Coefficient Matrix of $T(p(t))=\int_0^t\int_0^yp(x)dxdy$ that maps $P_3\rightarrow P_5$?

The usual basis for $P_n$, of course, is given by $\left\{1,t,t^2,\cdots,t^n\right\}$. Why is the integrand a function of $x$? Does this matter for the purposes of constructing a change of basis ...