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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How to find a matrix $B$ and an invertible matrix $P$ such that this matrix $A$ is in Jordan Canonical Form?

I am working on the following exercise: Find a matrix B and an invertible matrix P such that $$A = \begin{bmatrix} 1 & -2 & 1 & 0 \\ 1 & -2 & 1 & 0 \\ 1 & -2 & 1 ...
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18 views

In a binary code, all coordinates partake in at least one non-information set

It is true that all non-MDS $(n,k)$ codes contains at least one $k$-sized coordinate subset that does not correspond to an information set (because all such subsets are information sets iff the code ...
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2answers
72 views

Intrinsic proof for $(I + AB)^{-1}A = A(I + BA)^{-1}$ by using Schur complements on matrix block elimination

Given $(I + B(I - AB)^{-1}A)$ to be inverse of $(I + BA)$, how could we derive that the following alternative form holds $(I + AB)^{-1}A = A(I + BA)^{-1}$. This is easy to verify(direct proof). ...
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30 views

How do I calculate $x_1, x_2, x_3$ if matrix $Ax=0$?

$A=\begin{pmatrix} 1 & b &3 \\ 0& y &-b \\ 1& b-2 &2b+3 \end{pmatrix}$ for $b=-2$. I get result that it is $\begin{pmatrix}1\\ 2/7\\ -1/7\end{pmatrix}$ but it says that ...
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49 views

Calculate a linear transformation with a specific kernel

I just want to make sure that what I'm doing is correct. Here's the question: Determine a linear transformation $T$: $\mathbb{R^3} \rightarrow \mathbb{R^2}$ with kernel $W$: $W$ = {$(x,y,z)$ ...
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38 views

Using span of T to find Jordan

Looking for help on all parts of the question below: Let $U$ be the complex vector space of polynomials of at most degree 6. Define $D, T: U \rightarrow U$ by $D(f) = f'$ and $T = D^2 + D^3$ Set $V ...
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1answer
29 views

Proving that S+T could be an epimorphism

Suppose that $S, T$: $U \rightarrow V$ are both epimorphisms. Is it necessary that S + T is an epimorphism? So I know that what I need to prove is that $I(S+T) = V$ is or isn't necessary... But ...
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1answer
23 views

Maximize triple product with respect to orthogonality contraint

I have the following problem: Suppose I have a plane $p$ defined by point $\vec{q}_1$ and normal vector $\vec{n}$. Also I have a line $g_2$, defined by point $\vec{q}_2$ and direction $\vec{l}_2$ ...
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1answer
46 views

Find 10 commuting $2\times 2$ matrices of the same order

Prove that there exists 10 distinct real $2\times 2$ matrices which are pairwise commuting and all of the same finite order. Here, the order of matrix A is the smallest integer $k > 0$ such that ...
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1answer
32 views

Derivation of an identity for matrix inversion

I am reading a machine learning textbook by Bishop, in appendix, it shows an identity as follows $(P^{-1} + B^T R^{-1}B)^{-1}B^T R^{-1} = PB^T(BPB^T + R)^{-1}$ I would like to know if this identity ...
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164 views

Is there a way to cut a an ellipsoid with a plane such that it gives an circle?

I'm trying to answer this In $\Bbb {R^3} $ consider the ellipsoid: $2x^2+3y^2+4z^2=1$ It exists a subspace of dimension 2 which intersection with the ellipsoid is a circle. Justify any answer. ...
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21 views

A confusion about the norm of the restriction of a linear mapping.

Let $\Bbb X$ be a Banach space, $T:\Bbb X\to \Bbb X$ be a linear map and $P:\Bbb X\to \Bbb X$ be a projection operator. Denote the closed subspace that is the range of $P$ by $\Bbb Y:=\mathcal R(P)$. ...
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1answer
14 views

Do automorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an automorphism, i.e. a bijective endomorphism. If $V$ is finite-dimensional, we know that the characteristic ...
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1answer
70 views

Easy way to get Determinant of 4 by 4 matrix

I have learned one way to get $4\times 4$ determinant. That is, divide a matrix $A$ by 4 part where each part is $2\times 2$ matrix: $$A = \left(\begin{array}{cc} B & C \\ D & E ...
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I have a answer to a question about trace. Is there an easier answer to this question?

Let $A\in M_n(\mathbb{C})$. Show that $$tr\left(\frac{A+A^*}{2}\right)\leq tr((A^*A)^{1/2}).$$ My answer: It is easy to see that $$tr\left(\frac{A+A^*}{2}\right)=\text{Re}(tr(A))\qquad and\qquad ...
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22 views

Correspondence between bilinear forms and linear operators.

Let $v$ be a finite set. We define $l(V)=\{f: f:V\rightarrow \mathbb{R}\}$, this is a vector space with the usual sum and scalar product. This vector space has an inner product: $u,v\in l(V)$ ...
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2answers
63 views

Dimension of the subspace of the polynomial ring over $\mathbb R$

Suppose $P_n =\{ f(x) \in \mathbb R[x] : \deg(f(x)) \leq n\}$ and $W = \{ p(x) \in P_n : p(x) = p(1-x) \}$. Find the dimension of subspace $W$. Firstly I am showing that $W$ is a ...
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9 views

Mixed derivatives of coordinates from two representations related by an orthogonal transformation

Given two orthonormal vector representations $\overline{Y}$ and $\overline{Q}$ of an $\mathbb R^n$ space that are related by an orthogonal transformation $\overline{Q} = ...
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1answer
21 views

projections and column space

If $A^2=A$, is a projection and $v_1$ and $v_2$ make up the column space (basis) of the span why does $Av=v$? Why does multiplying $A$ times a column space vector not change the column space?
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Existence of an $n\times n$ real matrix $A$ such that $A^2=-I$.

Let $A$ be a $n\times n$ real matrix $A$ such that $A^2=-I$. Such an $A$ cannot be, Orthogonal. Invertible. Skew-symmetric. Symmetric. Diagonalizable. I tried to figure out the answer by looking ...
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31 views

Can we say scaling matrix is necessarily diagonal?

Can we say scaling matrix is necessarily diagonal? According to wikipedia, yes According to this video, no $S$ is scaling along orthogonal directions according to this So, how to put them both ...
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758 views

Why is this determinant positive?

I have seen that the $k$ dimensional volume of an parallelepiped in $\mathbb{R}^n$, i.e. $P(v_1, \ldots, v_k) = \{t_1v_1 + \dotsb + t_kv_k : 0 \le t_i \le 1 \}$, is $\sqrt{\det(T^{\top}T)}$, where $T$ ...
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1answer
32 views

Intersection of subspaces by construction

Let be $S \subset V$, subspace of a vector space $V$ over the field $\mathbb{K}$, $dim(V)=n$, and, $dim(S)=k<n$. For every $r\in \mathbb{N},$ $1≤r≤n-k$. Prove that $S$ is the intersection of $m$ ...
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30 views

Matrix operation - half inverse?

I understand the inverse of a $2\times 2$ matrix: $$\begin{bmatrix}a&b\\c&d\end{bmatrix}^{-1} = \frac{1}{ad-bc}\,\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$$ But what is the operation, ...
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1answer
29 views

Find the bases for the eigenspaces of the matrix.

The question I have is to find the bases for the eigenspaces. I have already found the characteristic equation which is $(λ-1)^2=0$. I also found that λ=1 The matrix I'm using is {(1,0),(0,1)} ...
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1answer
19 views

Find the Transformation matrix from M(nxn) space to R

Hi, For this problem I am supposed to find the bases B and C of the source and target vector spaces V and W and write the matrix [T]b,c that represents T with respect to these basis. I understand ...
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1answer
23 views

Finding out vectors that screw up linearly independence when given a set

I want to Find the vector space spanned by $A =$ {$(1,1,0,1),(1,2,-1,1),(3,4,-1,3),(-1,-3,-2,-1)$} By definition it's all the linear combinations I can make with those 4 vectors, However I ...
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1answer
36 views

An eigenvector with irrational ratios of coordinates

Let $A \in \text{Mat}_{n\times n}(\mathbb{Z})$. Suppose there exists $u =\begin{pmatrix} u_{1}\\ u_{2}\\ \vdots \\u_{n}\end{pmatrix} \in \mathbb{R}^{n}$ such that $Au = u$ and $\frac{u_{i}}{u_{j}}$ is ...
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2answers
32 views

Diagonalization of Linear Transformations

Given the linear transformation $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined by $T(x,y,z) = (3x-5z, \frac{1}{5}x - y, x+y-2z)$, find a basis $B$ for $\mathbb{R}^3$ such that $[T]_B$ is diagonal. ...
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1answer
35 views

Tricky basis transformation

I have a linear mapping from one basis to another $ F: \mathbb R^{3} \rightarrow \mathbb R^{4}$ with the bases being: $$ \mathbf v_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \\ ...
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3answers
47 views

Linear algebra: Show that there exists vectors $c_i \in \mathbb R^n$ such that $A\cdot c_i=b_i$

Let $A\in \operatorname{Mat}_{m,n} (\mathbb R)$ be a real matrix of rank $r=n$. Let $(b_1,b_2,..,b_n)$ be an orthonormal basis for the column space $R(A)$ (terms. the scalar product) Show that for ...
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2answers
30 views

Do endomorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an endomorphism. If $V$ is finite-dimensional, we know that the characteristic polynomial $\chi_f$ has a zero ...
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2answers
32 views

Intersection of two column spaces

Let the matrices A and B be: $$A = \begin{bmatrix} 2 & 3 & 0 \\ 2 & 3 & 1 \\ 2 & 3 & 1 \end{bmatrix} and \space B = \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 ...
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1answer
29 views

Simplify vector equation $2\mathbf c - (\mathbf a + \mathbf b)\times(\mathbf a - \mathbf b)$

The unit vectors $\mathbf a$ and $\mathbf b$ are both perpendicular to a third unit vector $\mathbf c$. Additionally, a is at an angle of $\dfrac{\pi}4$ to b. Simplify the expression: $$2\mathbf c - ...
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1answer
33 views

Bases for space of polynomials

I'm facing an exercise to determine basis for some spaces of polynomials. Here they are Consider the space of polynomials of degree equal or less than 3 $U =${$p(t) \in \mathbb{R_3}[t]$ | ...
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1answer
35 views

projections, column space

Let $A$ be an $n \times n$ matrix and denote its columns (in order) by $v_1, v_2 \ldots , v_n$ a.) If $A$ is a projection, explain why $Av_i=v_i$ for all $i=1,...,n$. b.) If $v$ is in the column ...
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1answer
15 views

What do non-invariant and non-orthogonal eigenvectors of an orthogonal matrix look like?

Suppose two vectors $u, v \in \mathbb{R}^{n}$ are non-invariant non-orthogonal eigenvectors of an orthogonal $n \times n$ matrix $A$. What are the eigenvalues of $A$ associated to the eigenvectors $u$ ...
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If I have matrix A, what is difference between $det(A), det(A_n), det(A_{n+1})$? [on hold]

If I have matrix $A_n$, what is difference between $det(A_n), det(A_{n+1})$ and $det(A_{n+2})$? If somone wants to help and answer on question can that be with example?
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Understanding matrix representation of a linear map

Let $V$ be a vector space over a field $K$ with ordered basis $B$. If $F:V\rightarrow V$ is a linear mapping then $M_B^B(F)$ is called the matrix representation of $F$ with respect to $B$. Does this ...
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46 views

Nullspace and column space of invertible matrix

I want to show that the matrix $A$ $n\times n$ is invertible if and only if $N(A) = {0}$ and $C(A) = R^n$. So far, this is what I've got: Theorem: A is invertible $\implies N(A) = 0$ and $C(A) = 0$. ...
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31 views

Adjoint of a bounded linear operator is bounded

Suppose $X$ and $Y$ are normed spaces over $\mathbb{R}$ and suppose $T: X \rightarrow Y$ is a bounded linear map. I want to prove that the adjoint map $T^\star : Y^\star \rightarrow X^\star$ is ...
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71 views

How do I prove that $\det A_{n+2} = a \det A_{n+1} + b \det A_n$ for matrix $A$?

I have calculated: $\det A_1=2$, $\det A_2=3$, $\det A_3=4$, so I was putting some numbers in $\det A_{n+2} = a \det A_{n+1} + b\det A_n$ like $n=1$, $n=2$ ($\det$ $n\times n$ matrix) and get that ...
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25 views

Gram-Schmidt help

Let $\Omega = [−1, 1] \subset \mathbb{R}$ and consider $\Pi_2(\Omega)$ Given that $B = \{p_1, p_2, p_3\}$ is a basis of $\Pi_2(\Omega)$ and that $$\langle f | g\rangle = f(−1)g(−1) + f(0)g(0) + ...
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Proof $S$ is the intersection of $m$ subspace of $V$ of dimension $n-r$

Let $V$ be a $k$-space with $dim(V)=n$. Let $S\subset V$ be a subspace, $\dim\left(S\right)=k<n $. For each $r\in\mathbb{N}$ with $1\leq r\leq n-k$, prove that $S$ is the intersection of $m$ ...
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53 views

Generators for a matrix group

Lets denote $\Gamma_0(4)$ the subgroup of $SL_2(\mathbb Z)$ : $$\Gamma_0(4):=\left\{\begin{pmatrix} a &b\\ c&d \end{pmatrix}\in SL_2(\mathbb Z), \ 4\mid c\right\}.$$ We also define $A$ and ...
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3answers
72 views

$A^{2014}=0$ for a matrix A

Let A be a 3*3 matrix and $A^{2014}=0$. Must $A^3$ be the zero matrix? I can work out that I-A is invertible, but I don't know how to proceed further.
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1answer
35 views

About a particular linear map between sequence spaces

Let $x \in \ell^1$ and $z \in \ell^2$ taking values in $\mathbb{R}$ and define a linear map $T_z: \ell^1 \rightarrow \ell^2$ as follows: $y_1=0$ and $y_n=\sum_{k=1}^{n-1}z_{n-k}x_k$ for $n\geq 2$. ...
2
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0answers
30 views

Linearity of Lebesgue measure

Suppose $\mu$ is the Lebesgue measure defined on $\Bbb R^k$, I want to show that $\mu$ has some kind of linearity, which seems intuitively correct: Suppose $A$ is a linear transformation on $\Bbb ...
2
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1answer
28 views

Can we have a diffeomorphism from a subset of $\Bbb R^2$ into a subset of $\Bbb R^3$?

In a lecture, our professor defined an allowable surface patch for a surface $S \subset \Bbb R^3$ to be a diffeomorphic surface patch of $S$. But is is possible to have a diffeomorphism between an ...
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17 views

Nomenclature for linear transformation

I was studying Linear Transformation and this symbol ↾ appears in this context: ST = S(T ↾ I(T)) ST is the composition of the linear transformations S and T I (T) image of T It also says after ...