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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Transformation matrix from principal angles and vectors

If I got it right, given two planes in $N$-dimensional space, their 2 principal angles ($\theta_1$, $\theta_2$) and 4 vectors ($\vec{a}_1$, $\vec{a}_2$, $\vec{b}_1$, $\vec{b}_2$) can be found, such ...
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11 views

Orthogonal projections exercise

Let $V$ be a $n-$dimensional space with inner product and consider $W$ a subspace of $V$. If $E$ it's a projections with $Im E = W$ such that $|E\alpha| \leq |\alpha|$ $\forall \alpha \in V$ then $E$ ...
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1answer
23 views

Basis for 4th degree polynomials such that integral of $p(x)$ from $-1$ to $1$ equals $0$

Let $U= \{ p \in \mathscr P_4\mathbb{R} \ | \int_{-1}^1 p(x)dx=0\}$. a.) Find a basis for $U$. b.) Find a subspace $W$ of $\mathscr{P_4}(\mathbb{R})$ such that $\mathscr{P_4}(\mathbb{R})= U \oplus ...
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22 views

In matrix algebra, what's the name for the inverse operation of pre- or post- multiplication?

For example, in this typical equation: $$\mathbf{Mv}-\lambda \mathbf{v}=\mathbf{0}$$ (where $\mathbf{M}$ is a symmetric matrix, $\mathbf{v}$ is a vector, $\lambda$ is a scalar, and $\mathbf{0}$ is a ...
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2answers
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Proving that $u$ and $v$ are linearly independent, given the independence of $T(u)$ and $T(v)$

Suppose that $T$ is a linear transformation and that $T(u)$ and $T(v)$ are linearly independent. Prove that $u$ and $v$ are linearly independent. I have no idea where to start in this case. Just need ...
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1answer
8 views

Finding all eigenvectors and eigenvalues of a linear operation on a function

Here is the question I am stuck on: Consider $T \in \mathrm{Hom}(\Bbb{R}[x]_{\le 2} ,\Bbb{R}[x]_{\le 2} )$ given by $$ (Tf)(x)=\int_{-1}^1(x-y)^2f(y)dy-2f(0)x^2$$ for all $f \in \Bbb{R}[x]_{\le ...
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1answer
28 views

linear algebra in infinite dimension

I look for an advanced linear algebra (A complete book but wich deals indiferently with infinite/finite vector space). To give an idea i expect a book that (for exemple) would prove the existence of a ...
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1answer
17 views

For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?

Is it true that for every integer $n>1$ , there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $A \in M(n,\mathbb R)$ and $AD=DA \implies A$ is also a diagonal matrix ?
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1answer
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Let $n>1$ and $g_1,…,g_{n-1}$ be $C^2$ scalar fields over $\mathbb R^n$ , then for any scalar field $f$ , is $\det J(f,g_1,…,g_{n-1})=0$?

Let $n>1$ and $g_i:\mathbb R^n \to \mathbb R$ be scalar field for each $1\le i\le n-1$ such that all second order partial derivatives of each $g_i$ exist and are continuous ( i.e. each $g_i$ is ...
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19 views

Derivative of $l_1$ norm

I want to compute the following derivative with respect to $n\times1$ vector $\mathbf x$. $$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 $$ My work: $$g = \left\lVert \mathbf x - A ...
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16 views

approximate projection into eigenvector space

Given a matrix A, $3 \times 3$, that is symmetric I calculate a matrix V, $3 \times 3$, whose columns are the corresponding right eigenvectors and a diagonal matrix D, $3 \times 3$. of eigenvalues so ...
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1answer
21 views

Computation of eigenvectors?

Given a matrix: $A = \begin{pmatrix} -\epsilon & tf_1 \\ tf_2 & -\epsilon \end{pmatrix}$ Compute the eigenvectors. I can easily find the eigenvalues to be $\lambda = -\epsilon \pm t\sqrt{f_1 ...
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1answer
39 views

Basis of the space of linear maps

I asked someone about this problem: Let $V,W$ be vector spaces with bases $(\alpha_i)_{i\in I}, (\beta_j)_{j \in J}$ respectively. Define $f_{ij}(\alpha_k) = \delta_{ik}\beta_j$. Show that ...
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1answer
19 views

Proof of a trace property

$Tr(XY) = 1$ and $Tr(Y) = 1$ implies that $Tr(X) = 1$. I tried to prove by contradiction and switch the dummy variable of $X$ and $Y$. But I don't think my approach is right and if there is any much ...
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1answer
15 views

Given a basis $U$, what conditions are needed for an orthogonal basis for it?

Given a basis $U$, what conditions are needed for an orthogonal basis for it? For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an ...
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34 views

If $\operatorname{rank}A=k$ then $A=A_1+…+A_k$ such that $\operatorname{rank}A_i=1$

Let $A\in M_n$ and $\operatorname{rank}A=k$. Is the following true? There are $A_i\in M_n$ ($i=1,...,k$), such that $\operatorname{rank}A_i=1$ and $A=A_1+....+A_k$.
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3answers
26 views

Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$

Let $N\in \text{Mat}(10 \times 10,\mathbb{C})$ be nilpotent. Furthermore let $\text{dim} \ker N =3 $, $\text{dim} \ker N^2=6$ and $\text{dim} \ker N^3=7$. What is the Jordan Normal Form? The only ...
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25 views

how to normalise these values

First of all, i don't know if the correct word is normalise or not, but I'll try to explain my issue. I have a relationship between an object A and an object ...
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25 views

What's the difference between the trajectory, the phase portrait and vector field of a matrix?

Take the matrix $$\begin{pmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{3}{4} & \frac{1}{4} \end{pmatrix}$$ as an example. What's the difference between its trajectory(discrete), phase portrait ...
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2answers
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matrix trace bilinear form

I'm wondering about this problem on bilinear forms : We have $\phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$ $$(A,B) \rightarrow trace(AB)$$ I've proved $\phi$ is a bilinear form ...
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$SU(n)$ generators

What is the generalization of the Pauli matrices and Dirac matrices in higher dimensions? I am actually looking for $\sqrt{\mathbb{I}}$ but I can't use the principal root which is just $\mathbb{I}$. ...
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1answer
18 views

Find orthonormal basis of quadratic form

Q: Let $$A = \begin{pmatrix} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{pmatrix}$$ Find the quadratic form of $q: \mathbb{R}^3 \to \mathbb{R}^3$ represented by A. and find ...
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27 views

Getting The Inverse Of A Positive Definite Matrix By Mutiplying It On A Diagonal One

Is the following true ? The inverse of a positive definite matrix is also positive definite and since symmetric then we could write the following: $A=PP^T, \space B = A^{-1} = P\Lambda P^T$ since ...
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1answer
33 views

If $V_1$ and $V_2$ are vector spaces over $\mathbb{Q}$ and f is a map $f(x+y)=f(x)+f(y)$ for all $x, y$ in $V_1$. Is $f$ linear transformation?

If $V_1$ and $V_2$ are vector spaces over the field of rational numbers and $f$ is a map from $V_1$ to $V_2$ such that $f(x+y)=f(x)+f(y)$ for all $x, y$ in $V_1$ show that $f$ is a linear ...
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10 views

Characters of Representations, Composition Series and Tensor Products

Let $(\pi, V)$ be a finite-dimensional representation of $G$. Prove the following: Suppose that $(\pi, V)$ has as a composition series $\{0\} \subset V_{1} \subset \dots \subset V_{r}=V$ with the ...
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2answers
28 views

Meaning of the phrase “span of vectors contains”

What does it mean if someone says the "span of vectors" $\{(x,y,z),(a,b,c)\}$ contains the vector $(d,e,f)$? I am making up numbers here. Let me know if I need to insert actual numbers here to ...
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2answers
47 views

Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$

I need some help figuring out how to work through this problem. Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$. Sorry, forgot to ...
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4answers
276 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = ...
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2answers
20 views

Minimize Energy Function

Let $A\in\mathbb{R}^{n\times n}$ be symmetric positive definite matrix and $\mathbf{b}\in\mathbb{R}^n$. How to prove that $A\mathbf{u}=\mathbf{b}$ if and only if $\mathbf{u}$ minimizes the so-called ...
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1answer
22 views

Existence of a block upper triangular form matrix representation for a linear operator

Let $T:V \rightarrow V$ be a linear operator on a finite dimensional vector space over $F$. Let $W \subset V$ be a subspace which is $T$-invariant. Show that there exists an ordered basis ...
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1answer
17 views

Is this the (a?) correct definition for $X$ having full row rank?

Let $X$ denote a $T\times K$ matrix. I have seen the definition for full column rank as "There is no vector $c \not = 0$ with $X\cdot c = 0$. Would a definition for full row rank then be "There does ...
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1answer
49 views

Solve this system of equations without calculator

$$2a +4b +3c +5d +6e=37$$ $$4a +8b +7c +5d +2e=74$$ $$-2a -4b +3c +4d -5e=20$$ $$a +2b +2c -d +2e=26$$ $$5a -10b +4c +6d +4e=24$$ find $a,b,c,d,e$ I tried solving the system of equations above but ...
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1answer
22 views

Is $X'X$ positive definite a necessary condition for $X'X$ to have full rank?

Let $X$ be a $T \times K$ matrix. $X'X$ positive definite means that for all $c \not = 0$ $c'X'X c >0$, so then $(Xc)' Xc > 0$ which implies $(X\cdot c)\cdot (X\cdot c)$ (I'm not sure what ...
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If all eigenvalues of A are zero then A must be similar to zero matrix. [on hold]

True or false. If true prove it else give an example.
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20 views

If $T$ is an orthogonally diagonalizable linear operator in an inner product space, show that $T^*$ is also orthogonally diagonalizable.

Suppose $V$ is an inner product space and $T$ is a linear operator that is orthogonally diagonalizable. Show that $T^*$ is also orthogonally diagonalizable. Here, $T^*$ denotes the adjoint ...
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1answer
18 views

Find upper triangular matrix C such that Cx=y

In the image above, how does one know that $c=e$ and $c$ is not equal to $f$? and $e$ is not equal to $f$? How does one know that $b=d$?
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Spectral norm of lower triangular perturbation

Suppose $A\in R^{n×n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. \begin{equation} A=I+L \end{equation} All diagonal entries of $L$ are equal to $0$, so that, ...
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Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...
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1answer
30 views

If $u$ is perpendicular to $v$ and $w$, then $u$ is perpendicular to $v + 2 w$?

True or false (give a reason if true or a counterexample if false): (a) If $u$ is perpendicular (in three dimensions) to $v$ and $w$, those vectors $v$ and $w$ are parallel. (b) If $u$ is ...
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$M_n$ is the subspace of all square matrices with trace $0$, what is the dimension of $M_n$?

There is an older post with many explanations of a more specific and less general case of a $4$ by $4$ Find the dimension of the space of $4\times 4$ real matrices with zero trace I didn't quite ...
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2answers
36 views

Linear transformation from a vector space to a field

Can anyone help me with the following question: Let $V$ be a vector space over field $F$, possibly not finite dimensional. Let $T \colon V \to F$ be a linear map. Prove that there is no subspace ...
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1answer
26 views

Showing that the following vectors are linearly independent in a subspace which they do not span.

I am trying to better understand vector spaces and dimensions. I could prove (i) via induction and the definition of linear independence? However how can I approach the questions (ii),(iii) which ...
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2answers
33 views

Row Switching Matrix

I know that there exists an elementary matrix that switches the rows in another matrix when they are multiplied, but how do you prove that this elementary matrix actually does this job? I am ...
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1answer
25 views

what are the smallest and largest values of $||v - w||$? What are the smallest and largest values of $v \cdot w$?

If $||v|| = 5$ and $||w|| = 3$, what are the smallest and largest values of $||v - w||$? What are the smallest and largest values of $v \cdot w$? How can I solve these two problems? For $||v - w||$ ...
2
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3answers
22 views

Proof: dimension of the vector space of solutions to the system Bx=0

I've run into a matrix dimension proof I'm having some trouble with: Let $A,B$ be $n\times n$ matrices, and let $P(A),P(B),P(AB)$ be the vector spaces of solutions to the systems ...
2
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1answer
20 views

Jordan canonical form in Lang's Algebra

In Lang's algebra on pp.559, he writes of the nilpotent part of a matrix $M$: "We observe also that the only case when the matrix $N$ is $0$ is when all the roots of the minimal polynomial have ...
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1answer
29 views

Show that Pn is an (n+1)-dimensional subspace [on hold]

Show that $P_n = \{$Polynomials with real coefficients of degree $≤ n\}$ is an $(n+1)$-dimensional subspace of the infinite-dimensional vector space of all real polynomials.
3
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2answers
31 views

Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
0
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1answer
20 views

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$.

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$. In my mind, a good way to go about this proof is proving that ...