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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Is set of three linear equations with three unknown solvable?

I have the following set of linear equations with the unknowns $h, n, i$ which I would like to express as a function of my known quantities, $e, f, g$: $$ e = h - n\\ f = h - i\\ g = i -n $$ with ...
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How can you find a matrix given you know its kernel/nullspace?

Suppose we are given that $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$, and also that $\ker\phi$ is the span of $\{\begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 ...
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Linear Operator with finite dimension

I'm involved with this exercise. I would greatly appreciate your help Let $V$ be a vector space of dimension $n$ over a field $F$. Let $T: V \rightarrow V$ a linear transformation whose image and ...
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Determining whether a set is linearly independent.

I am currently trying to determine whether the following set is linearly independent: $u=(4,3,-2), v=(2,-6,7), w=(14,-12,17)$ It can be easily observed that $w=2u+3v$ and since w can be expressed in ...
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Same eigen values giving 2 different eigen vectors

For the matrix below I am getting two eigen vectors for a single eigen value $$ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ ...
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How to determine whether a set is a vector space or not?

I'm currently learning Vector Spaces and although I understand the definition of what a vector space is, I can't seem to be able to find the correct answers when doing some questions. I would even say ...
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all abelian groups with 625 elements with 24 elements of order 5

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form $M = R/(p^{e_1}) \oplus \cdots \oplus R/(p^{e_t})$. Let $_pM = \{m \in M: p ...
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Singular values and trace?

Given that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?
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Injective linear mapping maps every plane to a plane through the origin?

Why an injective linear mapping from $R^3 \to R^3$ maps every plane to a plane through the origin? I can not understand this. It says also that if the mapping is not injective, it maps some line to a ...
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37 views

Matrix problem, subspace

Suppose you are given a matrix A and have calculated an echelon form R of A. (Note: R is not assumed to be in reduced row echelon form.) Which of the following statements must be true? (Select all ...
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28 views

Is inequality $tr(A^{-1^T} B) tr(A^T B^{-1}) \leq constant$ correct?

I have the following optimization problem \begin{align} \min_{A} &tr(A^{-1^T} B)\cr \text{subject to} &x^T A x > 0 \cr & A_{ii}=1 \end{align} where $A$ and $B$ are some positive ...
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Is the following set of vectors in $\Bbb R^3$ linearly dependent?

I am using Anton's Elementary Linear Algebra book (8e) and trying to do exercise set 5.3, question 2a It gives the vectors $(4,-1,2)$, $(-4,10,2)$ and asks if they are linearly dependent . My final ...
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21 views

For a given matrix $X$, find two linearly independent vectors in $C(X)^{\perp}$.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ Is there an easy way to ...
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38 views

Showing a $2\times2$ matrix is invertible

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that ${A} {w} = {v}.$ Show that ${A}$ is invertible. I have no idea on how ...
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Find an orthogonal basis for the space spanned by the columns of the given matrix.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ It is immediately clear to me ...
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32 views

To prove that on $C([0,1])$, the integral $\int_{0}^{1} f(x)g(x)dx $ defines a scalar product.

Now, for the given operation to be a scalar product, I know I need to check four conditions. Here's what I have done so far: $\langle f,g\rangle $ = $\int_{0}^{1} f(x)g(x)dx$ = $\int_{0}^{1} ...
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30 views

Find the matrix relative to the standard bases

Let $T:P_2 \to P_3$ be the linear transformation defined by $T(p(x)) = xp(x).$ Find the matrix for $T$ relative to the standard bases $B = \{u_1, u_2, u_3\}$, $B' = \{v_1, v_2, v_3, v_4\}$. $u_1 = ...
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49 views

Basis and dimension of the span of the vectors (0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)

Find a basis for the given subspace by deleting linearly dependent vectors. $S = \text{span}\{(0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)\}$ I do not understand how to "delete linearly independent ...
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34 views

Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. I know that $c$ is an egenvalue and $[1,...,1]^T$ is ...
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41 views

About summer course or online course of Linear algebra and real anyasis [on hold]

I just looking for the online course for Linear algebra or real analysis but it should be upper level. i saw MIT and another college but our university said it was not upper level its likely ...
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23 views

In $\mathscr{V}$, let $X \subset \mathscr{V}$ be a set of $n$ vectors. $Y \subset X$ contains vectors all scalar multiples, $X$ linearly dependent.

I would just like to verify that my proofs are sound and receive any suggestions on rewording. (If relevant, I am self-studying and haven't done a serious proof in about a year.) $\mathscr{V}$ is a ...
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Let $D$ be a nonsingular diagonal matrix. Show that $1\notin spec(DA)$ if and only if $D - A$ is nonsingular.

Let $F = \mathbb{F}_3$ and let $n$ be a positive integer. Let $D = [d_{ij} ]\in\mathscr{M}_{n×n}(F)$ be a nonsingular diagonal matrix and let $A\in\mathscr{M}_{n×n}(F)$. Show that ...
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34 views

Questions about Eigenspace

I'm learning about Eigenspaces and have a few questions. Do eigenspaces, eigenvalues, and eigenvectors correspond to a tranformation or can a single vector space $V$ have an eigen-stuff? Is an ...
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19 views

Reflect on y axis in 3D Matrix?

I have a question saying "Define a 3D Matrix that performs a reflection in the y axis" but I don't know how to solve it. So if we have a 2D matrix and we say 'reflection on the y axis' we mean that x ...
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43 views

Understand the definition of a vector subspace

I'm pretty new to Linear Algebra and I have started on Vector Spaces. I understand that a Vector space V over the set of real numbers is a set equipped with two operations, namely vector addition and ...
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Give two matrices whose column spaces contain the column space of the given matrix.

Let $$B = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\text{.}$$ Give two matrices whose column spaces contain $C(B)$, the column ...
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Relation between Ill-posed problem and eigenvectors?

This question is related to the question below: Is there a relation between Ill-posed problems and Eigenvectors. In the answer of the above question, it was shown that the ill-posed problem can be ...
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27 views

Dependence of linear algebra theorems of the commutativity of the field.

In the linear algebra course I took vector spaces where introduced with a (commutative) field. The classical theorems are proven under this assumption. However, I was wondering what implications it ...
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Orthogonal projection and subspaces

Consider the vector space $\mathbb{R}^m$ with usual inner product. Let $S_1$ and $S_2$ subspaces of $\mathbb{R}^m$ , $P_1\in\mathbb{M}_m(\mathbb{R})$ a orthogonal projection matrix on subspace $S_1$ ...
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11 views

Is there any relation ill-posed problem and not Normal matrix?

I am trying to understand different aspect associated with ill-posed problem. Can we claim that an ill-posed problem $Ax=b$ means that the matrix $A$ is not normal? Further, can we claim that if $A$ ...
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Orthogonal projection matrix proof

Let $P\in \mathbb{M}_m(\mathbb{R})$ a orthogonal projection matrix. Show that the matrix $Q=I-P$ is a orthogonal projection matrix. Make a geometric interpretation of the elements $z=Pb$ and ...
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Showing the action of $SO(p,q)$ in the punctured cone of isotropic vectors is transitive

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
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Show that a linear transformation $T$ is one-to-one

Problem: Consider the transformation $T : P_1 -> \Bbb R^2$, where $T(p(x)) = (p(0), p(1))$ for every polynomial $p(x) $ in $P_1$. Where $P_1$ is the vector space of all polynomials with degree less ...
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A Theorem on Skew-Symmetric Bilinear Forms in Hoffman and Kunze's Book

On pg. 377 in Hoffman and Kunze's Linear Algebra(Second Edition) Theorem 6 reads: Let $V$ be an $n$-dimensional vector space over a subfield of the field of complex numbers, and $f$ be a skew ...
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29 views

Finding a Basis for this subspace

Set $V=\mathbb{R}^{2x3}$ and let $U$ be a subspace of $V$ defined by: \begin{equation*} U=\{B=(b_{ij})\in V\mid b_{11} + b_{12} + b_{13} = -4(b_{21} + b_{22} + b_{23})\}. \end{equation*} I would ...
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1answer
68 views

Origin of the term dual space?

Basically, why is a dual vector space called as such? Is the reason for the term "dual" simply because the two vector spaces are related by a one-to-one mapping, or is there something more to it? ...
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1answer
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Expressing problems in canonical form for solving with simplex

The Picnic Hamper Company has a store containing 10,000kg of nuts, 4000 packs of smoked salmon, 2000 bottles of wine and 1500 Victoria sponges. It intends to use these goods to make up three different ...
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How do you find the value of n in this example

$$n^{n-2} = 16$$ I know $n = 4$ through trial and error but how do you find $n$ in a conventional manner? I'm basically trying to solve how many nodes are in a tree that has $16$ spanning trees ...
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Do all n x n matrices over the reals represent linear transformations?

Do all $v \in M_n (\mathbb{R})$ represent linear transformations? To add to that a bit to further clarify for myself: Looking up the def. of a transformation it is any function $f$ mapping a set $X$ ...
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Prove the surjectivity of this injective linear map

I am working on the following problem. Let $g : V\to V$ be linear and injective, where $V$ is a vector space over the field K. Prove that, if $V$ is finite-dimensional, then $g$ is surjective. In an ...
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Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
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Injective linear endomorphism of hilbert space is bijective?

Is it true that an injective continuous endomorphism of a hilbert space is bijective? If not, are there conditions that imply this? I know this would follow from the rank nullity theorem in finite ...
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1answer
7 views

Basis for row space of matrix: REF vs. RREF.

When finding a basis for the row space of a matrix, I reduce the matrix to row echelon form, and find the rows that have pivots in them. Does it matter wether you use the echelon or the reduced ...
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2answers
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$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
2
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1answer
25 views

Proving a matrix $A$ is of certain form

Let $A\in M_n(\mathbb{C})$, and $A=A^3$, prove that $A^2$ is of form $\begin{pmatrix} I_r & 0\\ 0 & 0 \end{pmatrix}$ where $1\leq r\leq n$. It make sense. My initial thought was to say that ...
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sum of matrices with unique solutions

Let $K$ be any field with a characteristic, different than 2, and $A$ any $n \times n$-matrix over $K$. For the equation $A = B + C$, where $A$ and $B$ are $n \times n$-matrices over $K$, are $B = ...
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3answers
15 views

Completing an orthonormal basis of a plane to a basis for $\mathbb{R}^3$

I was asked to find an orthonormal basis for the plane $x + 2y +3z =0$. I found a regular basis, $(-2,1,0),(-3,0,1)$, and then performed the Gram-Schmidt process to find 2 orthonormal vectors that ...
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1answer
30 views

Why is the the double dual functor on finite-dimensional vector spaces naturally isomorphic to the identity?

$\require{AMScd}$ Note: I have already seen this question, which asks about a specific aspect of the construction - here I am trying to construct this functor and failing at a very different stage. ...
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59 views

A Linear Operator of Rank 1

Let $T$ be a linear operator with rank $1$ on a finite dimensional vector space $V$.Then Which of the following are true? 1)either $T$ is diagonalizable or $T$ is nilpotent. 2)$T$ is both ...
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Find the complex eigenvectors, knowing the eigenvalues

If $$A= \begin{pmatrix}1 & -1 \\ h^2 & 1\end{pmatrix},$$ I know the complex eigenvalues are $1+ih$ and $1-ih$. How do we find the complex eigenvectors? Can someone please explicitly show me ...