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 ...

learn more… | top users | synonyms

0
votes
0answers
4 views

Plackett-Burman designs for screening experiments

i am looking for a way to investigate the relations between deferent factors in a experiment. basically i have 4 deferent lysis buffers, for each buffer i use deferent variables high/low amount of ...
1
vote
3answers
23 views

Tell if $A$ is diagonalized using it's characteristic and minimal polynomials

$$A= \left( {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right)$$ I already calculated that $f_A(x) = (x-3)(x-1)$. Also, the minimal polynomial must be also $(x-3)(x-1)$. How can I use ...
0
votes
1answer
16 views

Prove $A$ is scalar matrix

Let $A\in M_n(F)$ and let's assume $A$ has only one eigenvalue. Also, $A$ is diagonalized. Prove that $A$ is a scalar matrix. My Try: $${P^{ - 1}}AP = \left( {\matrix{ \lambda & {} & 0 ...
-1
votes
0answers
18 views

The basis for orthogonal complement of a subspace [on hold]

The following is my problem, thank you so much.
0
votes
0answers
10 views

Bilinear form Isomorphsim

Hello I'm trying to give a proof that the following are isomorphic: Bilinar forms and $ T_2^{0}(V) $ Where $ T_0^{2} = F \otimes V^{*} \otimes V^{*} $ and V vector space over F
0
votes
2answers
11 views

Proving that $\chi_{T^*}=\overline{\chi_T}$ and $m_{T^*}=\overline{m_T}$ (characteristic and minimal polynomials of adjoint map)?

For a linear map $T:V\to V$ where $V$ is a finite dimensional inner product space over $\mathbb{C}$, I know the result $\chi_{T^*}=\overline{\chi_T}$ (where $T^*$ is the adjoint map for $T$). My ...
0
votes
2answers
6 views

corresponding system of equation of the given solution space

The following question seems to me interesting. it gives solution space and required the corresponding system of equation. The question is the following: Consider the vectors in $R^4$ defined by ...
1
vote
0answers
19 views

How to applied Gaussian Elimination for non-full rank matrix

I have a question about gaussian elimination. I want to find solution of $$Ax=b$$ as soon as possible using Gaussian Elimination. This is my matrix A ...
1
vote
1answer
23 views

Example of matrices with some interesting properties like same characteristic and minimal polynomial etc.

Looking for two matrices $A$ and $B$ with entries in the field $F_2$ with the following properties: $A$ and $B$ both are invertible,have same minimal polynomial,Characteristic polynomial,same ...
0
votes
1answer
23 views

Some Dense subset of $M_2(\mathbb{R})$ with its usual topology?

The set of all invertible matrices i.e $GL_2(\mathbb{R})$ The set of all matrcies having both real eigen values. Having $Trace(A)=0$ $3$ is not dense set as It is closed set! $1$ Is dense. take ...
2
votes
1answer
23 views

Definition of angle between vectors in spaces with dimensions n

I am working on a problem which requires me to find certain values of the components of vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^4$ such that the angle between them is $\pi/3$ If my ...
2
votes
1answer
26 views

Analogy of transpose for a function?

In the page 2 of Linear algebra explained in four pages reference, it has a box describing the relationship between functions and linear transformation. It states that the set of zeroes of a function ...
2
votes
0answers
24 views

Does Least squares solution exist for this case?

$ {\bf{Z}} = {\bf{H}} \cdot {\bf{S}} + {\bf{N}} $ Dimensions of the matrices are as follows: Z = m X m H = m X n S = n X m (matrix S is an orthogonal matrix) N = m X m. All the elements of the ...
0
votes
1answer
24 views

Looking for proof of “ two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues ”

For any real square matrix $X$ let $P(X)$ denote the no. of its positive eigenvalues counting multiplicity . Let $A$ be a real symmetric $n \times n$ matrix and $B$ be a real invertible $n \times ...
1
vote
2answers
32 views

Complex projections order in inner product

So the complex projection is defined as $$\operatorname{proj}_\vec{u} \vec{v} = \frac{\langle \vec{v},\vec{u}\rangle}{\langle\vec{u},\vec{u}\rangle} \vec{u}$$ with complex inner product. I was ...
0
votes
1answer
34 views

What exactly is Standard Coordinates?

What exactly is a standard coordinates? Sorry it seems like a very stupid question, but my professor didn't really explain it and just started to use it for solving other problems related to ...
1
vote
2answers
32 views

establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$

establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$. I couldn't understand how to solve it please just give me the first step, maybe I can figure out the ...
1
vote
1answer
25 views

If $v\not=0$, then $v/\|v\|$ has norm 1

The question is: Show that if $\vec v$ is a non-zero vector in $\mathbb R^n$ then $\left( \dfrac{1}{||\vec v||} \right ) \vec v$ has norm $1$. I assume that $\vec v=(v_1,v_2,v_3,...,v_n)$ , ...
0
votes
1answer
21 views

Verify if symmetric matrices form a subspace

I need to verify if the symmetric matrices form a subspace. But I don't know how to represent a general symmetric matrix. I know that the matrix $A$ is symmetric if $A = A^t$ but I can't write a ...
0
votes
1answer
32 views

Obscure Rotation Matrix

The points of the shape after the shear are $(-0.25, 1)$, $(1.75, 1)$, $(0.25, -1)$, $(-1.75, -1)$. Other than that the only other information given is that the vertical edges of the original shaded ...
0
votes
1answer
28 views

$C$ & $D$ stuck at calculating

Hello! I am super stuck about how to calculate $C$ and $D$ in the example image, I know it's something simple I just can't figure it out!
2
votes
1answer
53 views

How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
0
votes
1answer
29 views

$r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
0
votes
0answers
11 views

Matrix coordinate blocks and null space of a vector

Suppose we have some non-zero $c \in \mathbb{R}^n$. Let $\{U_i\}_{i=1}^m$ be a collection of $n \times n_i$ matrices where each column is a standard basis vector. Suppose that each standard basis ...
0
votes
1answer
39 views

$A^{T}b$ inconsistent system!

I am trying to figure out how the calculation on the last image comes to be (question 9, the yellow area). I have calculated the rest without issue. I know that the formula for the last set is ...
0
votes
0answers
13 views

Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
0
votes
2answers
16 views

If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
0
votes
0answers
25 views

A reduction of Cayley-Hamilton to the complex case [on hold]

I found an interesting proof of the general Cayley-Hamilton theorem but i didn't quite get all of it. I'm trying to make sense of this reduction argument. I'm not that familiar with algebras so this ...
1
vote
1answer
15 views

Find a basis where vector has fixed cooridnates

What's the method for finding basis where $\beta_1=(0,2,1) \ , \ \beta_2=(1,1,2)$ and $\beta_1$ has cooridnates $1,2,-1$ and $\beta_2$ has $0,0,1$ ? we have that $(1,1,2)=(a_3,b_3,c_3)$ and ...
-1
votes
0answers
30 views

A linear system for which the solution space is spanned by the given vectors [duplicate]

Make a system with $3$ equations and $3$ unknowns of which the solution space $V$ is spanned by the column vectors: $$\begin{bmatrix} 1 \\0 \\-1\end{bmatrix},\quad \begin{bmatrix}1 \\3\\ ...
0
votes
0answers
23 views

What is recommended for studying Linear Algebra? [duplicate]

I don't know anything about linear algebra and want to start afresh but in a proper mathematical manner. What should I do and which are recommended? I know the basics of multi-variable calculus, some ...
1
vote
0answers
24 views

Find a family of linear inequalities for which the unit ball is a solution set

Find a family of linear inequalities for which the unit ball $A = \{x \in \mathbb R^n \ | \ \|x\|_2 \le 1\}$ is a solution set Would it just be $x_1^2 + ... + x_n^2 \le 1$?
2
votes
2answers
129 views

Algebraic operations with tensor products and direct sums.

When first learning about tensor products and direct sums of vector spaces, the analogy of multiplication and addition of vector spaces is sometimes used to help with the intuition. If we look at the ...
3
votes
0answers
20 views

Help with an Analysis problem involving isomorphisms and volumes.

I would like some help solving the following problem: Let $f: U \subseteq \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ be a class $C^{1}$ function. Suppose that for some $a \in U$ the derivative of $f$ ...
0
votes
1answer
28 views

prove linear independence of polynomials

Let $f_1,...,f_k$ be polynomials at field $K$ not including the zero polynomial and $\deg f_i \neq \deg f_j$ for every $i \neq j$ Show that polynomials $f_1,...,f_k $ are linear independent. I don't ...
0
votes
1answer
46 views

A Challenge on linear functional and bounding property

I took a midterm exam and after that wrote this problem down. My instructor was unable to solve it. The problem is copied here in order for anyone to help me. Suppose $f:E\to \mathbb{C}$ is a ...
1
vote
2answers
32 views

Find all 2 x 2 skew-symmetric matrices A [on hold]

Find all $2 \times 2$ skew-symmetric matrices $A$, if any, such that $A^2 + I_2 = 0$ Please help me! Thanks!!
1
vote
1answer
33 views

Give a general formula in terms of $n$ for the determinant of the following matrix.

Let $M_n$ denote the $n$ x $n$ matrix over $\mathbb{R}$ of which the entry in the $i$-th row and the $j$-th column equals $1$ if $|i-j|\leq 1$ and $0$ otherwise. For example: $M_6=$ \begin{pmatrix} ...
0
votes
0answers
11 views

Vectors transformation without using a linear regression or a neural network

Is there any way to do a regression (transforming a set of high dimension vectors ($dim=400$, $number=500$) from a state $A$ to a state $B$) without using a linear regression or a neural network. I ...
0
votes
1answer
19 views

Prove convexity of function over space of positive definite matrices

I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. What I have done: It seems like this problem could be tackled by considering the ...
0
votes
0answers
14 views

Algebraic expression of a regression matrix

Let's say I'm doing a multivariate regression between a set of input $n$-dimension vectors (noted by the matrix $X=\{X_1,X_2,...X_m\}$) and a transformed version (noted by the matrix ...
0
votes
1answer
29 views

For $n>1$ $\hat x$, $p$ & $P$ [on hold]

For $n = 1$, $$\hat x = \frac{a^Tb}{a^Ta}, \quad p = a\frac{a^Tb}{a^T a}, \quad P = \frac{aa^T}{a^Ta}$$ What are these formulas when $N>1$?
4
votes
4answers
98 views

$Ax=b$ what does solving it mean?

We have been going through how to solve the system of equations known as $Ax=b$. Where $A$ is a matrix, $x$ is a vector and $b$ is a vector. I understand that if we have $A$ and $b$ we must find out ...
0
votes
2answers
30 views

be J be a matrix so: JJ^-1 = I and A a matrix so: A^t JA = J. prove that A is invertible so that AA^-1 = I

be J be a matrix so: $JJ^{-1}$ = I and A a matrix so: $A^tJA^ = J$. prove that A can invertible so that $AA^{-1} = I$ the big question here is: what are the properties of A transpose, that allows ...
1
vote
1answer
21 views

A is normal if and only if every matrix unitarily equivalent to A is normal

I need to prove that if $A$ and $B$ are unitarily equivalent, then $A$ is normal if and only if $B$ is normal. The proof is as follows: Suppose $A$ is normal and $B = U^*AU$, where $U$ is unitary. ...
2
votes
1answer
50 views

How to find the basis?

So if there is a n-dimensional space (x_1, x_2, ..., x_n) over a field, How can I find a basis which together with (c_1, c_2, ..., c_n) (where all c_i's are non-zero constants from this field) forms a ...
0
votes
1answer
35 views

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ ...
1
vote
1answer
23 views

Proving or disproving the existence of a vector space

Suppose the vector space $V$ is spanned by $(1,0,1,0), (0,1,0,1), (1,1,0,0)$. Is it possible to find a subspace U of $\mathbb R^4$ such that $V \subsetneq U \subsetneq \mathbb R^4$? Note that ...
1
vote
1answer
32 views

Derivation of Simple Projectile Motion with Drag

Given the initial velocity $v_0$ and angle $\theta$ of a projectile on the ground, using Newton's second law and the acceleration due to gravity $\mathbf g=\left\langle0,-g\right\rangle$, I was able ...
1
vote
2answers
14 views

iid Gaussian random matrix $A\in M_n$ has full rank with probability 1?

I want to prove that: iid Gaussian random matrix $A\in M_n$(I mean whose elements are iid Gaussian) has full rank with probability 1 Below is my consideration: $$1-P(\text{full ...