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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What is the difference between a linearly independent set and a set that spans $\Bbb{R}^m$?

This is more of a conceptual question. Here's what I know about a linearly independent set of vectors: A set of vectors $\{v_1, ..., v_p\}$ is linearly independent if the equation $$x_1v_1 + x_2v_2 + ...
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1answer
20 views

How do I show that if T*T = Id$_{V}$ then TT* = Id$_{V}$, where T is a linear transformation and T* is its adjoint operator?

We can use that (T*)*=T because I have shown that while trying to work on this one. I am just plane stuck on this one. V is also finite dimensional
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0answers
24 views

Smallest distance of a point to a surface

Let $P$ be a hyperplane of dimension $n-1$ in the space $\mathbf{R}^n$, given some integer $n\ge 3$ (let's call the first axes $x,y,z,\ldots$). Then, fix a point $A \in P$ and define the surface ...
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0answers
15 views

Proving a linear operator is onto

The original question comes in two parts. The first part is: Let T be a linear operator on V. If $||T(x)|| = ||x||$, then T is one-to-one. I have done it as follows: For all x $\in$ $kernel(T)$, then ...
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2answers
44 views

What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...
1
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1answer
42 views

How to you find out what a matrix does to an equation.

Lets say I have an equation of a plane, $$x-3y+2z=0 $$ and I get matrix to transform it with say a 3x3 matrix with just a-i as place holders for the values in the matrix. How would I find what the ...
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1answer
16 views

Bounds on a quadratic form

I am currently in the middle of a proof where it would be nice to have some estimates on the size of a quadratic form. In particular, I am looking at $$x^TAx$$ where $A$ is "small" (in the analyst's ...
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0answers
14 views

Singular values of the differentation operator

I was trying to solve this exercise but I can't get to answer given in book: Find the singular values of the operator D over $R_2[x]$ (That is polynomials with degree equal or less than 2) define as ...
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0answers
26 views

What is the dimension of $\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[x]\}$?

I have $$\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[X]\}.$$ Here $f$ is the following endomorphism $$f(P) = (x^2-x+1)P(-1)+(x^3-x)P(0)+(x^3+x^2+1)P(1),$$ where $P\in\mathbb{R}_{n}[x]$. My ...
2
votes
1answer
60 views

$A$ and $B$ are positive matrices and $\|A+B\| = \|A\|+\|B\|$, then A and B have a common eigenvector

Suppose $A$ and $B$ are positive matrices such that $\|A+B\| = \|A\|+\|B\|$. Show that $A$ and $B$ have a common eigenvector, where $\|A\|$ is the operator norm of $A$. I'm wondering if there is a ...
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1answer
28 views

How can I write a set of equations in summation form?

I have a system of equations as follows: \begin{align} & A_1^{11} + A_1^{12} + A_1^{13} + \cdots + A_1^{1n}=X \\[8pt] & A_1^{21} + A_1^{22} + A_1^{23} + \cdots+ A_1^{2n}=X \\[8pt] & ...
0
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1answer
23 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
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1answer
18 views

Lower bound for the distance between matrices of different rank.

This is a follow up question to this: Norm of diference of matrices of different rank Suppose $A$ is a norm one $n\times n$ matrix of rank $k$. What is $\inf\|A-B\|$, where the infimum is taken over ...
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0answers
28 views

Completing the Square of Quadratic Forms

I was working through a proof of a lemma that lets us determine whether a Hessian is positive definite for Mardens' Vector Calculus, page 175 Basically the lemma is if $B= \begin{bmatrix} a ...
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0answers
23 views

Gram-Schmidt Process, finding orthonormal basis

Suppose I'm given $2$ random vectors $(v_1,v_2)$ I want to find orthonormal basis $(w_1,w_2)$ Are the following equivalent? for the $w_2$ case $$w_1=\frac{v_1}{\|v_1\|}$$ ...
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2answers
52 views

Diagonalizing, Eigenvalues and Eigenspaces

Prove that the matrix $A= \begin{pmatrix} 2 & 0 & -2 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix} $$ $ is diagonalizable and thus find the ...
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votes
1answer
67 views

If $B$ is the inverse of $A^2$, show that $AB$ is the inverse of $A$

I know that a question that is very similar, other than the wording, was asked here, but I am more interested in being critiqued on my proof. I would like to know if it is correct, if it is complete, ...
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1answer
14 views

Use the Gram-Schmidt procedure to construct orthonormal bases for the subspaces of Rn spanned by the following set of vectors

For part c: How can I quickly tell that the dimension of the subspace is 2? I used the algorithm and got "3" basis vectors before realising that the 3rd one was parallel to one of the others and ...
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1answer
14 views

Find the bases of the vector space of terminal sequences

Let V be the vector space of the sequences $ a = (a_0 , a_1 , a_2 , ...) $ of real numbers who are terminally - finally zero sequences (There is $ N $ such that $ a_n = 0 $ for every $ n > N $ ). ...
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votes
2answers
16 views

If $au + bv + cw = 0$ with $a+b + c = 0$ then $u,v,w$ are collinear

If $u,v,w \in \mathbb R^3$ such that for some $a,b,c$ real numbers with $a+b+c = 0$ we have $au + bv + cw = 0$, then why are $u,v,w$ collinear points? i substituted $a = -b-c$ and tried other things ...
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1answer
35 views

Is it unitary matrix or not? [on hold]

$A = \begin{bmatrix} \frac{i}{3^{1/2}} & \frac{1+i}{3^{1/2}} & 0\\ \frac{-1}{2^{1/2}} & 0 & \frac{i}{2^{1/2}}\\ \frac{1-i}{3^{1/2}} & \frac{1}{3^{1/2}} & 0 \end{bmatrix}$ Is ...
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0answers
4 views

$W=H^{\dagger}H=V diag(\lambda_1,\lambda_2,…,\lambda_N) V^{\dagger}$: Connection between $\textbf{H}$ and $\textbf{V}$

Let $\textbf{H}\in{C}^{M\times N}$ be a complex-valued matrix. And let $\textbf{W}=\textbf{H}^{\dagger}\textbf{H}\in C^{N\times N}$ with $\{\cdot\}^{\dagger}$ being transpose conjugate operator. By ...
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0answers
3 views

$Rad_{L}(f) \neq Rad_{R}(f)$

Give an example of bilinear form on a vector space $V$ such that $Rad_{L}(f) \neq Rad_{R}(f)$ We know that Let $V,W$ be a vector spaces of $f \in B(V,W; \mathbb{F})$. The left radical of $f$ consists ...
0
votes
1answer
19 views

Columns of a matrix linearly independent and spans

Let $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n \in \Bbb R^n$ and let $P$ be the $n\times n$ matrix whose columns are $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n$ I'm wondering why the followings are ...
2
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0answers
25 views

Generating $\mathrm{SL}(n,\mathbb{R})$

I have this question which doesn't seem too difficult and I would like to know if there is an elementary way to deal with it. I consider a closed subgroup $G$ of $\mathrm{SL}(n,\mathbb{R})$ which ...
0
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1answer
27 views

If you add row $1$ of $A$ to row $2$ to get $B$, how do you find ${ B }^{ -1 }$ from ${ A}^{ -1 }$?

If you add row $1$ of $A$ to row $2$ to get $B$, how do you find ${ B }^{ -1 }$ from ${ A}^{ -1 }$? Notice the order. The inverse of $B=\begin{bmatrix} 1 & 0 \\ 1 & 1 ...
2
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0answers
25 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A ...
-1
votes
0answers
17 views

Confusion of intersection of two 2-d planes in 4-d [on hold]

I've only just started a linear algebra course at my uni and I'm wondering if it is intuitive to say that two 2-d planes can't intersect in 4-d in such a way that they produce a 3-dimensional solution ...
0
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1answer
24 views

How to determine if (1,0,1,1), (1,1,0,1) , (0,1,1,1) spans $R^4$?

I set up a system where $a(1,0,1,1) + b(1,1,0,1) + c(0,1,1,1) = (1,1,1,1)$ (the standard basis of R4) then i found that $a + b = 1$ $b + c = 1$ $a + b + c = 1$ which implies that $a = c = 0,$ and ...
2
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0answers
14 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...
0
votes
4answers
55 views

Prove that a product of two complex numbers has zero imaginary part

This is my homework, which reads as follows: Let $z_1, z_2$ be complex numbers. Prove that when $z_1z_2 \neq -1$ and $|z_1| = |z_2| = 1$, then the imaginary part of $$ \frac{z_1 + z_2}{1 + z_1z_2} $$ ...
0
votes
1answer
92 views

For these subsets $S$, are they subspace for the indicated vector space $V$

Q1. $V =P_5(R)$ and $S=\{p(x)\mid p(15)=0\}$. I think it is a subspace, but not 100% sure. I tried let $p_1(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, such that $p_1(15)=0$ ...
1
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0answers
19 views

Finite field and its element with symbols [Sage / Python / …]

I have a finite field $T=GF(2^3)$, normal basis $(a, a^2, a^4)$ and polynomial $f$ from field $T$, which contains unknown variables / symbols. Is it possible to get vector with coordinates of f in ...
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0answers
18 views

Addition of Kronecker Product Matrices

Summary: Is it possible to write $A_1 \otimes A_2 + B_1 \otimes B_2$ as some object which has nice properties again, preferably as a Kronecker product itself? Each of the matrices $A_i$, $B_i$ can ...
2
votes
1answer
48 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
-1
votes
1answer
39 views

True or false? (linear algebra) [on hold]

If $u$ and $v$ are two solutions of $Ax$ = $0$, then any vector in $Span(u, v)$ is also a solution of $Ax = 0$. I have doubts with $span(u,v)$. I do not know if the same thing $span(u\cup v)$ or ...
0
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0answers
23 views

bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
0
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1answer
21 views

$Y$ coordinate of a point that lies on a line [on hold]

Given two points $A$ and $B$, for example $A(1,5),\,B(15,2)$, what is the $y$ coordinate of a point $C(x,y)$ lying on the straight line $AB$?
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0answers
59 views

In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
-1
votes
1answer
23 views

finding eigenvalue and eigenbasis

How do I find the eigenvalue and eigenbasis of $$T(f(x))= f(4)x$$ Theres more to the equation but i just need to figure out how to apply $f(4)$ as a base in figuring out how to do the rest of the ...
3
votes
3answers
27 views

multiplication of finite sum (inner product space)

I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book) Why need to be different index, i.e. $m,n$ when multiplying the two sums? ...
3
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0answers
21 views

Compact-open topology on $\operatorname{Hom}_\mathbb{R}(V,W)$

Suppose $V$ and $W$ are finite-dimensional real vector spaces, and I give $\operatorname{Hom}_{\mathbb{R}}(V,W) \cong V^* \otimes W$ its usual vector space topology. Does this agree with the subspace ...
1
vote
2answers
25 views

Norm of diference of matrices of different rank

Suppose $A$ is a $n\times n$ matrix of rank $k$ that has Euclidean norm equal to $1$. Given $p<k$, and $\epsilon>0$, can we always find a norm one matrix $B$ of rank $p$ such that ...
2
votes
0answers
12 views

Where can I learn properties about spaces of linearly independent projectors?

I am interested in characterizing the space of all collections of $d^2$ linearly independent projectors on the Hilbert space $\mathbb{C}^d$. The linear independence I desire is in the vector space of ...
0
votes
1answer
19 views

Systems generators that are not linearly independent

Good evening, I would find sets that are generators of vector spaces, but they are not linearly independent, ie they are generating space but are not a basis for it. For example for these spaces: ...
2
votes
1answer
27 views

Prove a semi-positive operator $T$ is an isometry if and only if $T$ is the identity operator.

Prove a semi-positive operator $T$ is an isometry if and only if $T$ is the identity operator. I was thinking that semi-positive means if $T$ is self-adjoint ($T^{*}=T$) and $\langle T(u),u\rangle ...
1
vote
0answers
53 views

Cramer's rule doesn't work here?

I tried to solve the following system: $$A_2\cdot 2\mathrm{i}\sin( \beta a) = B_3\exp(- \alpha a)$$ $$\mathrm{i} \beta A_2 2\cos( \beta a) = - \alpha B_3\exp(- \alpha a)$$ Then I got $A_2=0 ...
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0answers
14 views

How do I set the vector $\vec{v} = (0, 0, 1, 0, 0)^T$ as $\vec{\vec{v}}=\vec{u}+\vec{w}$ with $\vec{u}\in U$ and $ \vec{w}\in U^\perp$? [on hold]

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. enter image description here
0
votes
0answers
24 views

Let $V = R^3$ and let $U$ be the subspace spanned by $A= \{(-3,-2,0),(4,-1,2)\}$. Is there a subspace $W$ of $V$ such that the following holds?

Let $V = \mathbb R^3$ and let $U$ be the subspace spanned by $A= \{(-3,-2,0),(4,-1,2)\}$. Is there a subspace $W$ of $V$ such that the following holds? $$W \nsubseteq U$$ ...
-1
votes
2answers
21 views

How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension of $U^\perp$? [on hold]

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension ...