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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Problem in Linear Algebra about dimension of vector space

Let U and V be finite dimensional vector spaces Over $\mathbb R$, Let $L(U,V)$ be the vector space of linear transformations from $U$ to $V$, and Let $W$ be a vector subspace of $U$. If $Z$= {$T$ ...
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linear transformations of same form

Let $m$ and $n$ be positive integers and $\mathbb{F}$ be a field . Let $f_1 , . . . , f_n$ be linear functionals on $\mathbb{F}^n$ . For any element $a$ in $\mathbb{F}^n$ ,define : $$T(a) = ( f_1(a) ...
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11 views

Projectivity maps that fix three points

Please check this definition: A projectivity is a bijection $PV\to PW$ induced by an isomophism $\phi: V\to W$ given by $\phi(kv)=k\phi (v)$. Now, i have seen here Old Question that an answer says ...
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1answer
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Prove that linear functions on the space of polynomial functions is a basis for $V^{*}.$

Here is one of those examples from Linear Algebra by Kenneth Hoffman, Ray Kunze(p. 100): Let $V$ be the vector space of all polynomial functions from $R$ into $R$ which have degree less than or ...
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2answers
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How to find the orthogonal projection of the given vector on the given subspace $W$ of the inner product space $V$.

$V=P(R)$ with the inner product $\langle f(x),g(x) \rangle$=$\int_0^1 f(t)g(t )dt$, $h(x)=4+3x-2x^2$ and $W=P_1(R)={\{1,x}\}$. I don't know how to do this the question. All I know is that it has ...
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Isomorphisms between endomorphism algebras

Assume that $R$ and $S$ are associative $\mathbb{C}$-algebras with unit $1_R$ and $1_S$, respectively. In addition, assume that $_RM$ is a simple left $R$-module and $_SN$ is a simple left $S$-module. ...
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2answers
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Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
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1answer
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Find a basis for the following set of vectors

I'm unsure of how to find a basis for the set of vectors $\left(x,y,z\right)$ in $\mathbb{R}^3$ with $z=2x-5y$ I understand that you can prove vectors form a basis by showing they are linearly ...
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1answer
22 views

Prove the image of basis elements is linearly independent

I was wondering if someone could give me a quick proof or counterexample to the following statement. Let $f:V \rightarrow W$ be a linear map between finite dimensional vector spaces $V$ and $W$, both ...
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1answer
11 views

Is there a general formula for the $n$'th variable of the solution for a lower triangular linear system of equations?

I have a countably infinite linear system of equations $Ax = b$, where $A$ is lower triangular with $-1$ at all diagonal entries, and $b = \{-1/2,0,0,...,0\}^T$. I.e the $n$'th unknown depends solely ...
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Matrices for transformations

How can I find the matrices of part b and c? the answers are:
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General form (or parametrizations) of symmetric 3x3 matrices with repeated eigenvalues

I'm looking for the general form of a symmetric $3\times3$ matrix $\mathbf{A}$ with only two different eigenvalues, i.e. of a matrix with the diagonalized form ...
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2answers
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$A^2+B^2=AB$ and $BA-AB$ is non-singular

The question is: Are there square matrices $A,B$ over $\mathbb{C}$ s.t. $A^2+B^2=AB$ and $BA-AB$ is non-singular? From $A^2+B^2=AB$ one could obtain $A^3+B^3=0$. Can we get something from this? ...
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1answer
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Finding a matrix of a linear map with respect to Bases.

Let $f:U \rightarrow V$ be a linear map where $U$ and $V$ are finite-dimensional vector spaces. Let $U$ be the vector space of polynomials degree $3$ in variable $t$. $F:U \rightarrow \mathbb{R}$ be a ...
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Solving Three equations for 3 Unknowns

Today I have a question and I am really curious to know about this. Question: $$ 16y+39z+50zy=0$$ $$ 85x-78z+95zx=0$$ $$ 85x+32y+70xy=0$$ $$\text{Are The Equations like these can be solve for ...
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How to find the matrix representation $[T]_S,_B$

$B=\{x-x^2,1+x,1-x^2\}$ is a basis for $P_2$. Also, define T: $P_2\to P_2$ by $$T(a_0+a_1x+a_2x^2)=-a_0-a_2+(a_0+a_1)x+a_1x^2$$ My question is how to find the matrix representation of $[T]_S,_B$ ...
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Is the general equation for a straight line not considered a linear function in linear algebra?

Is the general equation for a straight line, which we called a linear function in highschool, i.e. $$f(x)=mx+c \tag{1}$$ not considered to be a linear function according to the linear algebra ...
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2answers
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Find the orthogonal projection of the given vector on the given subspace $W$ of the inner product space $V$.

$V=\mathbb R^2,u=(2,6), $ and $W={\{(x,y):y=4x}\}$. I've no idea about how to get through this. Please help in understanding this in detail,if possible pictorial representation will be best.
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Approximate spectral decomposition

See attempt below I am interested in effective and constructive computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a Hermitian ...
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nondegenerate bilinear form $\mathrm{dim}{S}+\mathrm{dim}{S^{\perp}}=n$ [duplicate]

I was told that in a linear space $V$ with nondegenerate bilinear form$\langle\cdot,\cdot\rangle$ , and $S$ is a subspace of $V$. we have $$ \mathrm{dim}{(S)}+\mathrm{dim}{(S^{\perp})}=n $$ where ...
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Non-degenerate symmetric bilinear form; dimension formulae.

Let $E$ be a vector space endowed with a non-degenerate symmetric bilinear form. Show $\dim F+\dim F^{\perp}=\dim E=\dim\left(F+F^{\perp}\right)+\dim\left(F\cap F^{\perp}\right)$ Lang uses this ...
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+50

Does anyone know a reference to best-fitting lines with integral coefficients?

I'm writing up a manual on how to generate "nice" Linear Algebra problems; that is, where the solutions tend to be integral. I "discovered" the following fact about the best-fitting line: Theorem. ...
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1answer
2k views

Simultaneous diagonalization of two positive semi-definite matrices

Let matrices $A, B$ be two $n \times n $ positive semi-definite matrices ; they can be represented in the following form $$A=\sum_{i=1}^{n} \psi_{i}p_{i}p_{i}^{T}=P\Psi P^{T}, \quad B=\sum_{i=1}^{n} ...
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1answer
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What does it mean to say an ordered pair of vectors of random variables is independent of one anotehr

Suppose we are given i.i.d. random observations $\{ y_i,{\bf x_i} \}$ where $y$ is scalar and $\bf x$ is a vector. When one say $\{ y_i,{\bf x_i} \}$ and $\{ y_j,{\bf x_j} \}$ are independent of one ...
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1answer
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Find the singular decomposition of T i.e. orthonormal bases and scalars so $Tv=…$

Find the singular decomposition of $$T=\begin{bmatrix} 0\ 0\ 4 \\ \frac{5}{2}\ \frac{-1}{2} \ 0 \\ \frac{-1}{2}\ \frac{5}{2} \ 0 \end{bmatrix}$$. That is find an orthonormal basis $(e_1,e_2,e_3)$, ...
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1answer
23 views

Transformation of Orthonormal Bases

Suppose that $u_1, . . . , u_n$ and $v_1, . . . , v_n$ are orthonormal bases for $\Bbb{R}^n$. Construct the matrix A that transforms each $u_i$ into $v_i$ to give $Av_1 = u_1, . . . Av_n = u_n$.
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Find the matrix of the operator $f(T)$ relative to the ordered base of $V$

Let $V$ be a vector space of finite dimension over the field $\mathbb F$. $T$ is an operator on $V$: $T:V \to V$. $B$ is an ordered basis of $V$. The matrix $T$ relative to the basis $B$ is $A$. If ...
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Suppose that $V$ is an inner-product space and $T \in {\scr L}(V)$. Prove that if $\|T^{*}v\| \leq \|Tv\|$ for all $v$, then $T$ is normal. [on hold]

Suppose that $V$ is inner-product space and $T \in {\scr L}(V)$. Prove that if $$\|T^{*}v\| \leq \|Tv\|$$ for all $v \in V$, then $T$ is normal.
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A Variant of Diagonally Dominant Matrix

I know that a matrix $\mathbf{A}=[a_{i,j}]$ where $|{a_{i,i}}|\geq\sum_{i\neq{j}}|{a_{i,j}}|, \forall{i}$, is called diagonally dominant. but what do you call a matrix $\mathbf{A}$ if: ...
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Finding latitude and longitude of point using radio direction [on hold]

I am working on a project where there are radio receivers. When these receivers get a signal they transmit their location (in latitude and longitude) and the heading (in degrees) towards the signal. ...
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1answer
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Find unique rotation matrix

Given some unit vector $v \in R^3$, consider the set of linear transformations $S$ such that if $L \in S$, $L(e_3) = v$ The matrix of $L$ has determinant 1 There is no rotation about $e_3$ I ...
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True or false (and explain!): if $W$ is a subspace and $x$ is a vector, then $Proj_Wx$ is orthogonal to every vector in $W$. [on hold]

True or false (and explain!): if $W$ is a subspace and $x$ is a vector, then the projection of vector $x$ onto subspace $W$ is orthogonal to every vector in $W$. I think this is true because the ...
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1answer
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If $L$ is a diagonalizable linear operator, why is $f(L)$ well-defined for $f: \mathbb{C} \rightarrow \mathbb{C}$?

In a book on quantum mechanics, I encountered a statement equivalent to the following. Suppose $V$ is a finite dimensional inner product space over $\mathbb{C}$. Let $L : V \rightarrow V$ be a normal ...
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1answer
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Orthongonally diagonalize the matrix A

Orthongonally diagonalize the matrix A, by finding an orthongonal matrix Q, and a diagonal mathrix D, such that $Q^{t}AQ=D$ $$A=\begin{bmatrix} 5 & 6 & 0 \\ 6 & 5 & 8 \\ 0 & ...
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1answer
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Does the set of all 3x3 echelon form matrices with elements in R form a subspace of M3x3(R)? Same question for reduced echelon form matrices.

Screenshot of the past exam question Firstly, the zero 3x3 matrix denoted as A is both in echelon and reduced echelon form since it satisfies both definitions respectively. $$ A = ...
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4x5 linear equation treaded as parameter

I got a 4x5 linear equation (4 equation 5 incognitas)like this: 1 1 0 0 0 = 800 0 1-1 1 0 = 300 0 0 0 1 1 = 500 1 0 0 0 1 = 600 i tried to give solution taking ...
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2answers
1k views

Linear Algebra and planes in Cartesian space

I was asked this question from the course Linear Algebra and I need to show all working. The question is in 5 parts: Consider the xyz-space R3 with the origin O. Let l be the line given by the ...
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1answer
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Square root of a matrix as it relates to the identity

Prove that for any $2×2$ matrix $M$ which is “sufficiently close” to the identity matrix, there exists a matrix A such that $A^2 = M$, and that this matrix A is unique if $A$ isrequired to be ...
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Let $A$ be a $4\times4$ matrix with real entries and eigenvalues $1$, $-1$, $2$ and $-2$, then which of the following statements are true?

If $B$ is a matrix defined as $B=A^4-5A^2+5I$, where $I$ is a $4\times4$ identity matrix, then $\det(A+B)=0$ $\det(B)=0$ $\operatorname{tr}(A+B)=4$ From the given conditions, I could only ...
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1answer
448 views

Using Gauss elimination to check for linear dependence

I have been trying to establish if certain vectors are linearly dependent and have become confused (in many ways). when inputting the vectors into my augmented matrix should they be done as columns or ...
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1answer
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Theorem, Inverse of Matrix is the Sum of Power

I noticed that $(1+S)^{-1}=\sum_{n=0}^{\infty}S^n$, where S is a square matrix. Is there any theorem related this identity?
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Dual spaces and dual basis [on hold]

If V is a finite dimensional vector space over the field F with dual space V* = Hom(V,F) . How to prove every ordered basis for V* is the dual basis for some basis for V ?
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Understanding singular value decomposition

Please, would someone be so kind and explain what exactly happens when Singular Value Decomposition is applied on a matrix? What are singular values, left singular, and right singular vectors? I know ...
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1answer
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Computing change of base matrix

I'm having trouble understanding how to solve the following exercise (or rather, what is it asking for): Find the change of basis matrix for the following basis B and D for $\mathbb{R}^2$. ...
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Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$

Let $\cal A$ be the (noncommutative) unitary $\mathbb Z$-algebra defined by three generators $a,b,c$ and four relations $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$. Is it true that $ab\neq 0$ in $A$ ? This ...
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1answer
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Let A be a $3\times3$ real orthogonal matrix. Prove that there exists a vector $w$ in $R^{3}$ such that $Aw=w$

Let A be a $3\times3$ real orthogonal matrix. Prove that there exists a vector w in $R^{3}$ such that $Aw=w$ or $Aw=-w$ . Tried: $(Aw)'Aw = w'A'Aw=w'w$$\implies Aw=w$ I think it is incorrect.
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1answer
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Why is $\bar{B^t} \bar{A^t} = \overline{(AB)^t}$ true? [on hold]

In class I've encountered the following thing: $\bar{B^t} \bar{A^t} = \overline{(AB)^t}$. I don't understand why that's true.
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Find a Matrix $C$ that will reduce matrix $A$ to the diagonal form, $C^{-1}AC$

Find a Matrix $C$ that will reduce matrix $A$ to the diagonal form by the transformation $C^{-1}AC$. I know that we are able to diagonalize a matrix and set A = $PDP^{-1}$ where P is the eigenvector ...
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1answer
15 views

A family of vectors is linearly independent.

Let $K$ be a field and $E$ be a $K$-vector space of dimension $n$. Let $\phi$ be an endomorphism of $E$. Let $(\lambda_1,\cdots,\lambda_n)$ be a family of distinct scalars and $(x_1,\cdots,x_n)$ be ...
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Proving or disputing existence of required linear transformation

I came across the following question which seems fairly easy, but I'm afraid that I'm missing something. There is a linear transformation $T:\ R_3[x]\to\mathbb{C}$ where both spaces are over ...