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 ...

learn more… | top users | synonyms

4
votes
3answers
94 views

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
0
votes
1answer
41 views

If two linear functionals are such that the kernel of one is contained in the kernel of the other, then they are proportional [duplicate]

Let $V$ be a vector space over $K$ and let $f,g \in V^*$ and satisfy $\ker f \subseteq \ker g$. Show there exist such $c \in K$ so that $c \cdot f =g$ How to approach this problem ?
0
votes
2answers
9 views

Interchange rows in a matrix without using interchange operation

I'm sure that it's already out there somewhere in the abyss that is page 37 on google, so I apologize. I haven't been able to find it. Given some arbitrary matrix, how can two rows be interchanged ...
1
vote
0answers
15 views

How to find the orthogonal of a vector space

Let $V$ be a vector space over a field $F$ equipped with a symmetric bilinear form $B$. Let $W$ be a vector subspace of $V$. I know that we define the orthogonal complement $W^\bot$ to be ...
0
votes
0answers
17 views

Find basis of subspaces

I don't know how to create basis of V1 and V2. If I want to prove M1^2=M1, do I need to find matrix representation of M1 first? Thanks!!!!!!
0
votes
2answers
13 views

consider if given vectors are elements of the span?

Consider the vectors u = (1,3,2) and v = (2,-1,1) in ℝ³. Determine whether or not (1,7,5) ∈ span(u,v) . Not really sure what to do, I was thinking of checking to see if u and v span ...
0
votes
1answer
17 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
-2
votes
3answers
22 views

Show that a linear map $f:A\to B$ such that $\mathrm{dim}\,A> \mathrm{dim}\,B$ can't be 1-1.

Given a linear map $f:A\to B$ such that $A, B$ are vector spaces and $\mathrm{dim}\,A> \mathrm{dim}\,B$, show that $ \ f$ can't be 1-1.
0
votes
1answer
18 views

Kernel of $q(x,y,z)=2x^2-4xy+2z^2-4xz+4yz$

I have some problems when calculating the kernel of the quadratic form $q(x,y,z)=2x^2-4xy+2z^2-4xz+4yz$: indeed, I get $Ker=\{(x,y,z)|x^2-y^2=0\}$, which results in a 2-dimensional kernel. Could you ...
-2
votes
0answers
16 views

How to solve the vector differential equation?

I'm new to this section, so I'm trying to solve vector differential equations, and I need some guidance. Could anybody give a step-by-step process for doing so, so that I could do some more problems ...
1
vote
0answers
8 views

Conditions of invertibility, linear transformations

Please, I need a hint. :) Let $T:\Bbb R^m\rightarrow \Bbb R^n$ and $ U:\Bbb R^n \rightarrow \Bbb R^m $ be linear transformations. What are the conditions that $m, n$ have to satisfy to $UT:\Bbb R^m ...
1
vote
1answer
12 views

Matrix representation induced by quotient space

someone can help me with this question, I know how to solve ker(A) but I don't know how to develop matrix representation. Thanks!!!!!
0
votes
2answers
26 views

Sense of rotation. How would the rotation matrix look like for this “arbitrary” axis?

My first question is how do you define the sense of rotation about an arbitrary axis? Rotations are usually counterclockwise and when referring to rotation with respect to the $x$,$y$ or $z$ axis ...
1
vote
1answer
34 views

What is the derivative of (Ax)'

Let $f(x)=(Ax)^T$ where A is a matrix and x is a vector. How do you explain that $f'(x)=(Ax)^T$? Specifically, that $\frac{\partial}{\partial x} f(x) (y) = (Ay)^T$. I can't seem to do it rigorously. ...
1
vote
3answers
16 views

Find all linear operators such that $F^2 = F$ and $F(x,y) = (ax,bx+cy)$

I need to find all linear operators that match $F^2 = F$ and $F(x,y) = (ax,bx+cy)$ *where $F^2$ means $F$ composed with itself. So what I did: $F(x,y) = (ax,bx+cy)\implies F(F(x,y)) = ...
7
votes
4answers
790 views

Show that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$

I know about the fact that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$, where $A$ and $B$ are $m \times n$ matrices. But somehow, I don't find this as intuitive as ...
0
votes
1answer
297 views

Annihilators in dual space

Let $X$ be a real linear space, $X'$ be a set of all linear functional on $X$. For $V\subset X$, $L\subset X'$ we define: $$ V^\bot=\{f\in X': f(x)=0 \textrm{ for } x\in V \}, $$ $$ L_\bot=\{x\in X: ...
0
votes
1answer
14 views

Kernel of the quadratic form $q(x,y,z)=3x^2+6xy+10xz+2yz+3z^2$

Let $q:\mathbb{R^3} \to \mathbb{R}$ such that $$q(x,y,z)=3x^2+6xy+10xz+2yz+3z^2.$$ I have to determine rank, signature, kernel, and the canonic form of q (with its matrix). I have solved most of the ...
0
votes
0answers
7 views

How to Simplify/Rewrite this Expression into a Generalized Eigenvalue Problem - via Similarity perhaps?

I have the following optimization problem: \begin{eqnarray} min~b' y' Z (Z' \Omega Z)^{-1} Z' y b \end{eqnarray} such that $b'b=1$. The matrices are $Z \in R^{n \times k}$, matrices $y \in R^{n \times ...
1
vote
1answer
39 views

Help with finding basis of a vector space

Let $A = \left[\begin{array}{cc} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \\ \end{array}\right]$ and vector $u = \left[\begin{array}{cc} 2 \\ 1 \\ 1 \\ \end{array}\right]$ ...
-1
votes
0answers
17 views

How to construct transformation matrix between two group basis?

If I have one group basis $A=\{e_1,e_2,\cdot\cdot\cdot,e_n\}$, and another group basis $B=\{e_1^{\prime},e_2^{\prime},\cdot\cdot\cdot,e_n^{\prime}\}$ of $\mathbb{R}^n$, I want to figure out the ...
17
votes
9answers
25k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
0
votes
0answers
27 views

Lower and Upper Triangular Matrices

$A$ is an $n\times n$ matrix and $L$ is an $n \times n$ nonsingular lower triangular matrix. How can I prove that if $LA$ is lower triangular, then $A$ is lower triangular? How can I do the same for ...
0
votes
1answer
11 views

Intersection of “positive” open half-spaces

Prove that the intersection of "positive" open half-spaces associated with any basis $x_1,x_2, \ldots, x_n$ of a finite dimensional vector space $V$ is non-empty. Recall that the "positive" open ...
0
votes
3answers
72 views

$U(R) \equiv U(B) \equiv U(N)$, trying to find unique values that result in indifference.

I'm currently working on a decision problem, and for some reason I am struggling with a system of equations, which should be the easiest part of the problem. The correct answers are ...
2
votes
2answers
28 views

Rank of $ T_1T_2$

For n$\ne $ m let $ T_1 :R^n \to R^m $ and $ T_2:R^m\to R^n $ be linear transformations s.t $ T_1T_2 $ is bijective. Find rank of $ T_1$ and$ T_2$. I tried by fact that bcoz $ T_1T_2 $ is bijective ...
1
vote
1answer
21 views

Why is a linear autonomous system asymptotically stable iff for all eigenvalues $\lambda$ of $A$, $Re(\lambda) < 0$

I'm trying to understand asymptotic stability of linear antonymous systems. I'm not sure if for the system $x' = Ax$, $x(t) = 0$ is the only fixed point that can be stable. In any case, I can ...
-3
votes
1answer
27 views

What is true for rank of a $5\times5$ matrix [on hold]

Let $A$ be a $5\times 5$ matrix and let $B$ be obtained by changing one element of $A$. Let $r$ and $s$ be the ranks of $A$ and $B$, respectfully. Which of the following statements is/are correct: ...
0
votes
1answer
16 views

How to find the base of Hom(U, V)

I know that Hom(U, V) is a vector space, which means it has a base. What is the way to find the base of it?
1
vote
0answers
27 views

Linear algebra of state space representation won't be linear (superposition theorem)…

After answering a question about calculating the state space representation of a circuit with 3 sources in it (the circuit is there), I had a doubt - while checking, it became clear there is something ...
0
votes
0answers
17 views

Elementary Matrices, Replacing a row

Lets say I have a 4 x 4 Matrix and I want to replace one row of that matrix with a different row from the same matrix. I need a matrix that when multiplied to the original matrix achieves this. I ...
1
vote
1answer
37 views

In $\mathbb{R}^3$, if $v$ is orthogonal to $x$ and $y$, then $x \times y$ is a scalar multiple of $v$. [on hold]

Let $x, y, v \in \mathbb{R}^3$. If $v\neq0$ is orthogonal to $x$ and $y$, then $x \times y$ is a scalar multiple of $v$. We can do $$v\times(x\times y)=(v\cdot y)x-(v\cdot x)y=0$$ so, by ...
1
vote
1answer
34 views

Vector Question… Stupid vector question.

Find a non-zero vector $\textbf u$ with terminal point $Q(-3,2,0)$. such that $\textbf u$ has the same direction as $\textbf v =$$ (3,1,-2)$ Since vectors are defined by their components and not ...
2
votes
4answers
125 views

Definition of basis

There are something that I am not quite sure about the definition of basis. Let $V$ be a vector space over $K$, then the definition of basis says the vectors $v_1,...v_n$ form a basis of $V$ if they ...
0
votes
1answer
35 views

Find a basis for $\mathbb{R} ^5$ containing the given vectors

Find a basis of $\mathbb{R}^5$ that contains the vectors $(1,-1,1,-1,0)$, $(-1,-1,1,-1,0)$ , $(-1,1,1,-1,0)$. I think I need to find two more vectors so that the five vectors are all linearly ...
2
votes
1answer
57 views

determinant of matrix $X$

Please hint me. ‎How ‎can I ‎calculate ‎determinant ‎of ‎matrix ‎‎$‎X‎$‎?‎ \begin{equation*}‎ ‎\mathbf{X}=\left(‎ \begin{array}{ccc}‎ A&B&‎\cdots&B\\‎ B&A&‎\cdots& B\\‎ \vdots ...
5
votes
1answer
87 views
+100

Intuition behind functional dependence

What is the intuition behind functional independence ? (This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally ...
0
votes
1answer
25 views

Orthogonally diagonalizing a matrix

Can anybody explain how to orthogonally diagonalize the following matrix: $$ \begin{pmatrix} 9 & \sqrt10 \\ \sqrt10 & 0 \\ \end{pmatrix} $$ Am I correct in ...
1
vote
2answers
27 views

Similarity in two 2x2 Matrices and finding the S in A=SBS-1

I am doing something wrong here and I am not sure what. The object of the exercise is to find the S for similar matrices $A$ and $B$. $A=SBS^{-1}$ with $B=\begin{pmatrix}4& 1\\1& ...
0
votes
1answer
18 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
0
votes
1answer
12 views

When does the Singular Value Decomposition fail?

Does the singular value decomposition ever not work? The statement of the associated theorem, here from wikipedia: http://en.wikipedia.org/wiki/Singular_value_decomposition#Statement_of_the_theorem is ...
-1
votes
0answers
22 views

Im and Ker of matrix

If Ker are the solutions of the homogenous system and Im are for the $A^t$ matrix what is the point of defining them ?What is their purpose?
0
votes
1answer
31 views

How to calculate the Matrix of a given Linear Transformation?

Let $V = F^3$ and $W = F^4$ and we define the following functions: $p\in {\cal L}(V,F)$ given by $p((x,y,z)) = 3x + 4y + 2z$ $q\in {\cal L}(W,F)$ given by $q((w,x,y,z)) = 2w + 5x + 7y + 11z$; $T\in ...
0
votes
0answers
25 views

Linear transformation of vector

I have computer graphics class and i had something like that on lecture: $$ \begin{bmatrix} \overrightarrow{b1} & \overrightarrow{b2} & \overrightarrow{b3} \end{bmatrix} \begin{bmatrix} c1\\ ...
0
votes
1answer
54 views

Differentiating a matrix function with respect to a scalar

I would like to differentiate the following with respect to psi (partial): $$ \operatorname{trace}\bigl((X^\top X)^{-\psi} P\bigr). $$ Here we have that: $ X \in \mathbb{R}^{p \times n}, P \in ...
-1
votes
2answers
31 views

Relationship between the four fundamental sub spaces

I am self studying linear algebra from Gilbert strang. I can understand the dimensions of the four subspaces but I am having trouble understanding the four subspaces from the perspective of linear ...
3
votes
1answer
38 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
0
votes
1answer
22 views

Prove that $b_{\perp}^{T}b_{\parallel}=0$

If $A \in \mathbb{R}^{mxn}$ then the unique expansion of every $b \in \mathbb{R^{m}}$ is $b =b_{\perp}+b_{\parallel} $. Prove that $b_{\perp}^{T}b_{\parallel}$. Comment: Saying that they are ...
0
votes
1answer
21 views

Linear Transformations of Functions

$\textbf{Problem}$ Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = mx + b$. $\textbf{a.}$ Show that $f$ is a linear transformation when $b = 0$. $\textbf{b.}$ Find a property of linear ...
1
vote
1answer
307 views

The possible set of eigenvalues of a $4\times 4$ skew symmetric, orthogonal matrix

The possible set of eigenvalues of a $4\times 4$ Real skew symmetric, orthogonal matrix is $1.\{\pm i\}$ $2.\{\pm i,\pm 1\}$ $3.\{\pm 1\}$ $4.\{\pm i,0\}$ As it is real skew ...