Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Prove that the function T is a projection.

Let $T:V\rightarrow V$ be a projection on the vector space $V$. Prove that: $I - T$ is a projection $V = \ker(I-T) \oplus \mathrm{im}(I-T)$ How do I show that $(I - T)^2 = (I - T)$. I think $I$ ...
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Doubt on determinant and linear independence

I am confused about this matrix. We know Row rank = column rank = determinant rank for a matrix and proof is known to all. See the following matrix $$A = \left[\begin{array} &t &t^2\\ 0 ...
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612 views

Proof related to direct sum and subspaces

I did the following exercise: If $U_1, U_2, W$ are subspaces of $V$ with the property $V = U_1 \oplus W = U_2 \oplus W$ then $U_1 = U_2$. My proof: Assume $u_1 \in U_1 \subset V$. Then $u_1 \in U_2 ...
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56 views

Example that is not a subspace

I did some linear algebra exercise and did the following: Give an example of a nonempty subset $U$ of the xy-plane with the property that $U$ is closed with respect to scalar multiplication but $U$ is ...
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55 views

Show that $V$ is $T$-invariant.

Let $T:W \to W$ be a linear operator vector space $W$ over $\mathbb{F}$. such that $w \in W$ where $$\{w, T(w) ,T^2(w)\}$$ is linearly independent and $T^3(w)= w +T(w)+T^2(w)$. Show that $$V := ...
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$A$ complex, not diagonal, can $A^*A$ be diagonal?

If $A$ is a complex matrix which is not diagonal, can $A^*A$ be diagonal? My first impression is that it cannot, and my mind runs to the fact that $\operatorname{Tr}{(A^* A)}\geq \sum ...
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86 views

Proof by example?

I was doing some linear algebra exercies. One I did is this: Prove that the union of two subspaces $U,W$ of $V$ is a subspace of $V$ if and only if one contains the other. My proof is this: If one ...
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96 views

on the Singular Value Decomposition

If $T$ is a self-adjoint linear map on a $n$-dimensional inner product space $X$ (either real or complex) then we know by the spectral theorem that there is an orthonormal basis of $X$, call it $v = ...
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190 views

Positive semi-definite matrix problem

If $A,B$ and $M$ are positive semi-definite matrices, and we have $$ A+B \succeq M .$$ Do there always exist two positive semi-definite matrices $ M_{1}, M_{2} , $ such that $$ A \succeq M_{1}, ...
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645 views

What is the mathematical intuition behind àl-jàbrà?

The term algebra comes from the arabic term àl-jàbrà that means "to force", "to restore". Over centuries mathematicians, in east and west, celebrate by this term mathematical disciplines. What is ...
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154 views

Iwasawa Decomposition

I was asked to prove that if $$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$ denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ...
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97 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
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Is this a valid proof of why a linearly dependent matrix has a $0$ determinant?

If a $3\times{3}$ matrix has a non-trivial solution to $B\vec{x}=\vec{0}$, then it has linearly dependent rows. Looking at Cramer's rule: $$x_1=\frac{\begin{vmatrix}0 & b_{12} & b_{13} \\0 ...
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3answers
898 views

Why is $\det (A-\lambda I)=0$?

I'm not sure I understand the logic behind why $\det (A-\lambda I)=0$ for any non-trivial solution to $(A-\lambda I)x=0$.
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50 views

Determinant of symmetric matrix of the form $v\otimes v$

Note that for $V=\mathbf{R}^n$, $$S^2V = \{ v\otimes w \mid v, w\in V\text{ and }v\otimes w=w\otimes v \} =\{ A\in \mathrm{M}_2(\mathbf{R}) \mid A=A^T \}.$$ Clearly, $S^2V $ contains $O=\{ v\otimes v ...
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156 views

Ways to calculate the inverse of a matrix, assuming it exists…

I'm wondering - Other than by using row reduction on the augmented $[A|I]$ to get $[I|A^{-1}]$, and by reducing a matrix to a product of elementary matrices, is there any other way to determine what ...
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167 views

Vector space bases without axiom of choice

I want to find an example of a vector space with no base if we assume that axiom of choice is incorrect. This question might be duplicate so please alert me. Thanks.
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59 views

Exhibiting A Basis

Exhibit a basis and calculate the dimension of the following subspace S of $\mathbb{P}_2$. $S=\{a+b(x+x^2) \mid a,b \in \mathbb{R}\}$ $\mathbb{P}_n$ denotes the vector space with polynomials ...
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332 views

Closed Form for Continuant (Determinant Tridiagonal Matrix)

Consider the particular tridiagonal, $n \times n$ matrix $A$: A = $\left(\begin{array}{ccccccc} a_1&b_2 ...
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92 views

How to check linear independence of these vectors?

Let $R=k\langle x,y\rangle$ be the free algebra on two variables. We can think of it as an algebra of non-commutative polynomials. Consider elements of the form $p[x,y]q$, where $p,q\in R$ are some ...
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75 views

If $T: V \to W$ is an isomorphism, then there are bases $B$ and $B'$ such that $[T]_{B',B}$ is the identity matrix

Suppose $V$ and $W$ are finite-dimensional vector spaces and that $T: V \to W$ is an isomorphism. Then there exist bases $B$ and $B'$ for $V$ and $W$ respectively such that $[T]_{B',B}$ is the ...
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Apostol question on alternative definition of dot product

The problem says: Suppose we define the dot product by $A\cdot B = \sum_{k=1}^n |a_kb_k|$. Which of the following properties hold with this new defition? Does the Cauchy-Schwarz inequality still ...
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986 views

Hyperplanes and intersection

How to prove that if I have two hyperplanes in $\mathbb{R}^{n}$ that have only one point of intersection, then $n=2$ (or $n=1$ trivially).
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464 views

What's a good book on advanced linear algebra?

I'm taking an advanced linear algebra course and I'm a little confused about books. The teacher said we could use any book we wanted to, but he recomended just Hoffman and Kunze and also Kostrikin, ...
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Linearity of the determinant

I'd like to prove the following properties of the determinant map. $\det I = 1$ $\det$ is linear in the rows of the input matrix The determinant map is defined on $n\times n$ matrices $A$ by: ...
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1answer
31 views

Is the dual of an equivariant metric equivariant?

Let $g$ a finite dimensional $K$-vector space, and let $g:V \otimes V \to K$ be an inner-product. If As usual, we can use the musical isomorphisms of $g$ to define an inner product on $V^*$, which we ...
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507 views

Endomorphisms of a finite dimensional vector space

From Humphreys' Introduction to Lie Algebras and Representation Theory: If $V$ is a finite dimensional vector space over $F$, denote by $\text{End }V$ the set of linear transformations ...
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1answer
131 views

LU factorization with pivot to solve linear system

I read that LUP matrix exist for any square matrix such that it is not a singular one. But I came across a matrix that when I get the LUP and calculate Lz=Pb -> ...
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If $Af(B) = B$ and the constant term of $f$ is nonzero, $f(B)$ is invertible

I am trying to solve the following problem: Let $A,B \in M_n(\mathbb C)$ be matrices and $f\in \mathbb C[X]$ such that $Af(B) = B$. Prove that if $f(B)$ is not invertible, $f(0)=0$. I set out ...
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Proving a Set is NOT a vector space

Before I begin, I will emphasis I DO NOT want the full solution. I just want some hints. Show that the set $S=\{\textbf{x}\in \mathbb{R}^3: x_{1} \leq 0$ and $x_{2}\geq 0 \}$ with the usual rules for ...
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3answers
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Are parallel vectors always scalar multiple of each others?

I read this in a tutorial of a university course : We note that the vectors V, cV are parallel, and conversely, if two vectors are parallel (that is, they have the same direction), then one is ...
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60 views

How is row elimination getting rid of this entry?

This is a really elementary question, but I want to make sure I'm not missing something conceptual. In Strang's book Linear Algebra and Its Applications, on p. 321, he introduces tests for positive ...
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Are there open problems in Linear Algebra?

I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. There ...
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1answer
88 views

Does a “typical” unitary matrix have an entry of magnitude 1?

I guess that a "typical" unitary matrix (or "almost every" unitary matrix) in $d \geq 2$ dimensions does not have an entry with magnitude 1. I would like to make this statement more precise and see a ...
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40 views

linear map property.

I found the following theorem in "Friedberg-Lienar algebra 4ed". " Let $~~V,~W~$ be vector space over field F. Let $~ \varphi ~: V \to W ~~$ be $~~$ isomorphism . Then, For any $~Q \subset V$ ...
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110 views

Images in a short exact sequence

Suppose $$ 0\to V\to W\to X\to 0\\ \downarrow\quad\quad\downarrow\quad\quad\downarrow\\ 0\to V'\to W'\to X'\to 0\\ $$ is a commutative diagram of vector spaces, with the top and bottom rows short ...
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48 views

getting PDF from a given Moment Generating Function

if the moment generating function mgf of a random variable w is M(t)=(1-7t)-20 find the i)pdf ii)mean iii)variance of w
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82 views

The generalized eigenvectors of linear operator $T$ span space $V$, why?

I'm studying about determinant and I have a problem understanding the following (Proposition 3.4): The problems I have are highlighted with red rectangles. If anyone can, could you clarify these ...
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224 views

How to prove that a matrix $U$ is unitary, if and only if the columns form an orthonormal basis?

And also, is it true that a matrix is unitary if and only if $T^{-1}=T^{*}$ ? Thanks.
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1answer
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A question about the index of the matrix $A$

Let $A\in \mathbb{C}^{m\times n}$ . We say the nonnegative integer number $k$ to be the index of matrix A , if $k$ is the smallest nonnegative integer number such that $rank(A^{k+1}) = ...
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2answers
109 views

How to show a subspace is invariant with respect to a group action

Suppose we have two finite vector spaces A and B. Consider a finite group G that acts on A and B via linear transformations. Assume when G acts on A and B there exists only the trivial subspaces that ...
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Solving a vector equation

I have the equation: $\mathbf{k} \times \mathbf{A} = \mathbf{B}$. $\mathbf{k}$ and $\mathbf{B}$ are known and I need to find the components of A. As it stands this system is indeterminate. I choose ...
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49 views

Showing a particular map is equivariant with respect to certain group actions

Let $A$ = {triangles in $\mathbb{R^2}$}. We can let $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$ be the vertices of the triangle. The group $GL(2,\mathbb{R})$ acts on $A$ by acting on the vectors of the ...
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244 views

Isomorphisms between vector space subspaces

Originally, I was trying to to understand this proof from Axler: Proposition: If V and W are finite dimensional, then $\mathcal{L}$(V,W) is finite dimensional and dim $\mathcal{L}$(V,W) = (dim ...
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1answer
73 views

Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$

Is there a nice presentation for the group $\text{PSL}(2,\mathbb{F}_q)$, for every prime $q$? (I don't have any particular definition in mind for "nice", other than, say, a small number of generators ...
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293 views

Find the matrix $S$ of stretch by a factor of $3$.

All mappings are from $\mathbb{R}^2$ to $\mathbb{R}^2$. Find the matrix $S$ of stretch by a factor of $3$ in the $y$-direction and the matrix $S^{-1}$. So the matrix $S$ is a $2\times2$ matrix. So ...
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1answer
92 views

finding a equation to fit a curve

If I have a set of known values, i.e X Y 0.81300, 4.9900 0.84500, 3.6900 0.86400, 3.0700 0.94000, 1.5000 0.94300, 1.4600 How would I make as accurate a ...
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86 views

How to solve Aschbacher's exercise on symmetric forms

Aschbacher's Finite Group Theory is an excellent textbook, and its exercise 7.9.10 on page 104 is used to justify several assumptions in the chapter on classical groups. Let $F$ be a field and $f$ ...
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69 views

Are there $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus D$, and $A\cong C$, but $B\not\cong D$?

I remember an exercise from Roman's Linear Algebra, but now I can't locate it in the book. Anyway, I think it asked to give examples of $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus ...
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146 views

If $\langle Tv,v\rangle\in\mathbb{R}$, prove that $T$ is self-adjoint

Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, together with a Hermitian inner product $\langle\cdot\,,\cdot\rangle$. Let $T:V\to V$ be a linear function. Prove that $T$ is ...