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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Need to check the meaning of a Transition matrix

Is the transition matrix just the change of basis matrix from a non-standard basis to the standard basis?
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77 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
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75 views

Scalar Matrix and Diag Matrix

A set of all diagonal matrices (nxn) over R is a field relatively to additive and multiply operations on matrices. A set of all scalar matrices (nxn) over C is a field relatively to additive and ...
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34 views

Matrix inequality: conjugating positive matrix by $R<-I$

Consider a symmetric positive definite matrix $P$ and arbitrary matrix $R<-I$. Does the following inequality hold? $$ P < RPR^T $$ If yes, provide some references. If no, guide me under what ...
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162 views

Basis of M2,2 is not spanning set of trace zero matrices?

given set of matrices: S={[1 0; 0 0];[0 1; 0 0];[0 0; 1 0];[0 0; 0 1]} I have to explain why S is not a spanning set of matrices with trace zero, matrices of: V be the subspace of M2,2: V = {[a b; ...
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find adjoint of a linear transformation defined by a cross product

Suppose that a in R^3 is given. Define A in L(R^3) by Av = a (cross)v find adjoint A, assuming that R^3 is equipped with the standard dot product I know to find adjoint A, I must start from definition ...
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36 views

relationship between number of polynomials and dimension of the space.

If p1,p2,...,pk are linearly independent polynomials in Pn, a mathematical relationship between k and n is: k<=n. If the k will be more than n, the set of polynomials can not be linearly ...
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2answers
33 views

Normalization of product of matrix and vector

Let $\hat{\mathbf{v}}=\frac{\mathbf{v}}{||v||}$, $\hat{\mathbf{M}}=\frac{1}{\det \mathbf{M}}\mathbf{M}$ Do the entries of $\mathbf{\hat{M}\hat{v}}$ always lie between $0$ and $1$? I can see how all ...
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Strength of “Every finite dimensional subspace of a vector space has a complement”

Does the following choice principle have a name? Every finite dimensional subspace of a vector space has a complement. Equivalently, every line inside a vector space has a complementary ...
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2answers
131 views

Proving that irreducibility of a matrix implies strong connectedness of the graph [duplicate]

I have tried to prove that if a matrix $A\in\mathbb{C}^{n\times n}$ is such that there are no two sets $I,J\subseteq\{1,\dots,n\}$ that are disjoint, complementary, nonempty, and such that for all ...
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3answers
54 views

This is about matrix $m \times n$ as vector space

I got a question from my book, to prove $m \times n$ matrices with standard operation is a vector space. I know that I must prove 10 axioms of vector space, and I confuse in the inverse, that is $u + ...
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25 views

Let {$c_1$,$c_2$, , , , , $c_n$} and {$d_1$, $d_2$, , , , , $d_n$} be non-zero real numbers and let $a_{ij}$ = $c_id_j$ . how to find rank of $A$

Let {$c_1$,$c_2$, , , , , $c_n$} and {$d_1$, $d_2$, , , , , $d_n$} be non-zero real numbers and let $a_{ij}$ = $c_id_j$ . how do I find the rank of the matrix $A$ =$a_{ij}$ Somebody please give me ...
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34 views

Mixed Interprogramm remodeling

for example i have the following problem min z 5 x_1a + 6 x_1b - 3 x_2a + 0 x_2b <= z -3 x_1a + 0 x_1b - 1 x_2a + 2 x_2b <= z x_1a + x_1b = 1 (Constraint say of this group only one variable ...
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306 views

Matrix-Trace and Conjugate Transpose (Multiple Choice)

I was trying to solve the following problem from a competitive exam paper. Let $A=( a_{ij})$ be a nXn complex matrix and let $A^*$ denote the conjugate transpose of $A$. Then which of the following ...
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1answer
149 views

how to find an upper bound of the spectral radius

I've given the real valued matrix $K=K_1+K_2$, with $K_1$ and $K_2$ symmentric and positiv defined. Further there are given this 3 matrices: with $\omega > 0$ and Now I tried the whole day ...
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128 views

Determine all values of $\lambda$ with which vectors $2, x, x^\lambda,\ (\lambda\in R)$ are linearly dependent for every $x\in R$.

Vectors are linearly dependent if $\alpha_1v_1+\alpha_2v_2+...+\alpha_nv_n=0$ and at least one $\alpha_i\neq 0, \ i=\overline{1,n}$. So, using this fact I write $\alpha_12+\alpha_2x+\alpha_3 ...
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6answers
89 views

Any linearly independent set in a vector space is a basis for that space?

Any linearly independent set in a vector space is a basis for that space? Is that true or false in general? I would think it would be true because the fact that is it a linearly independent set would ...
4
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1answer
143 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
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2answers
285 views

Vector Space-Linear Transformation (Multiple Choice)

I was trying to solve the following problem from a competitive exam paper. Let $A$ be a nonzero linear transformation on a real vector space $V$ of dimension $n$. Let the subspace $V_0 \subset V$ be ...
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2answers
26 views

We have a system of linearly independent vectors: $x, y, z$. Determine, whether system $x, x+y, x + y + z$ is linearly dependent.

So, we have linearly independent vectors $x,y,z$, so for $x, x+y, x + y + z$ if I cannot express $x$ using the other two vectors (there is no way to express $x$ using $x+y$ and $x+y+z$, as far as I ...
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1answer
35 views

Localizing a point using distance measurements to four points in 3-D

This article explains how to do trilateration step by step. I need to extend this process to 3-D. As far as I know, I need four distance measurements in order to calculate a fifth point's coordinates. ...
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25 views

Basis of an implicitly given subspace

$ \left\{ t\pmatrix{7 \\ -3 \\ 0 \\ 1}+s\pmatrix{7 \\ -5 \\ 1 \\ 0},s,t \in R \right\} \in R^4$ Determine the basis of the subspace I would say the basis is $ \pmatrix{7 \\ -3 \\ 0 \\ ...
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35 views

proof that the vectors are linearly dependent

$v_1,v_2,w \in R^n$ $w $ is in the span {$v_1,v_2$} Show that the vectors {$v_1,v_2,w$} are linear dependent. My first approach was: $w=a*v_1+b*v_2$ and because $w$ is in the span of ...
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2answers
66 views

Proof: $\exists$ subspace $U$ of $ker(f)$ with $U \bigoplus T_1 = T_2 $

I need help with this proof: Let $V, W$ be K-vectorspaces. Let $T_1, T_2$ be subspaces of V with $T_1 \subseteq T_2$. Let $f \in hom_K(V,W)$. Show the following: If $ f(T_1) = f(T_2)$ then exists a ...
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1answer
66 views

Why is the intersection of spans zero?

As a part of a larger proof, my text claims that if $$A\begin{bmatrix}u_1&u_2\\ \end{bmatrix}=\begin{bmatrix}u_1&u_2\\ \end{bmatrix}\begin{bmatrix}\lambda&1\\0&\lambda\\ ...
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0answers
49 views

Reachable set using constant control input

Consider a linear time-invariant control system given by the differential equation \begin{align*} \dot{x}(t) &= Ax(t) + Bu(t), \;\; x(0) = x_0, \end{align*} where $x\in \mathbb{R}^n$ and $A,B$ ...
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154 views

Does adding two linear equations will result in a line which will pass through an intersection of the linear equations?

I was wondering why it is almost impossible to find a geometrical explanation of why adding two linear equations helps us to find a solution of a system of linear equations? Am I right that adding two ...
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31 views

About canonical linear programs.

Starting out with linear programming, I'm having some questions about canonical linear programs: Do all linear programs have a canonical form? So far I couldn't figure an example stating otherwise. ...
3
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1answer
78 views

Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
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1answer
59 views

Existence of invariant subspaces of $\mathbb C^n$ and $\mathbb R^n$

Exercise $6.15$ from Dym's Linear Algebra in Action: Let $A \in \mathbb R^{n\times n}$ and suppose that $n \geq 2$. Show that: $(1)$ There exists a one-dimensional subspace U of $\mathbb ...
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3answers
223 views

Choosing good textbooks in linear algebra, analysis and graph theory

I need some advices to choose good undergraduate textbooks in LINEAR ALGEBRA, ANALYSIS and GRAPH THEORY. I found: Gilbert Strang // Introduction to Linear Algebra - Welleslay Cambridge Press (2009) ...
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2answers
36 views

Base of a Subspace

$a,b\in R^n$ are base of the subspace U. is $(a+2b,b)$ a base of U? My guess was to show that $q*a+w*b=r(a+2b)+t(b)$, but how can I show that the formal way.
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1answer
40 views

Theorem: Euclidean space $\langle E;+\mid R\rangle$ is normed if $||x||=\sqrt{(x,x)}$ , $x\in E$.

How do I prove this theorem? Maybe Cauchy's inequality needs to be applied here?
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2answers
110 views

Proof that if $Ax=b$, then $x=A^{-1}b$

Let's say we have $Ax=b$ where $A$ is a matrix. What is the proof that if we multiply both sides of equation with a matrix (inverse of $A$ in this case), then they are still equal?
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1answer
74 views

Matrix multiplication makes zero to specific elements

I have this matrix $A = \pmatrix{3&2&5\\ 1&2&3\\ 2&3&5\\ 1&3&4}$. I want to multiply this matrix by another one, say $M$, which must have its diagonal elements = 1, and ...
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1answer
46 views

Finding an image (Linear Transformations) $T(2,-1,1)$

Let $T:R^3 \rightarrow R^3$ be a linear transformation such that $T(1,1,1) = (2,0,-1), T(0,-1,2) = (-3,2,-1)$ and $T(1,0,1)=(1,1,0)$. Find the indicated image $T(2,-1,1)$ I used the rule that: ...
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1answer
49 views

Functions on adjacency matrices

From Norman Biggs, Algebraic Graph theory 2j, p13: The adjacency matrix has a spectral decomposition $A = \sum \lambda_aE_a$, where the matrices $E_a$ are idempotent and mutually orthogonal. (...) It ...
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2answers
68 views

The inner product of cross products.

I need to prove: $$\langle a \times b | c \times p\rangle = \langle a|c\rangle \langle b|p\rangle - \langle a|p\rangle \langle b|c\rangle$$ $a,b,c,p$ belong to $M$ where $M$ is 3D real inner product ...
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1answer
74 views

Help determining whether a function is a linear transformation $T:M_{2,2}\rightarrow R, T(A)=|A|$

Again, here is the function: $T:M_{2,2}\rightarrow R, T(A)=|A|$ I was able to prove that its not a linear transformation because $T(A+B) \neq T(A)+T(B)$ in fact, $T(A+B) = C$ where $C$ is a new ...
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Show that AB is singular if A is singular

Actually I need to show that $\det(AB) = \det(A)\det(B)$ if $A$ is a singular matrix. The determinant of $A$ is $0$ if $A$ is singular, so $\det(AB)$ has to be $0$ as well, but I have problems ...
2
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1answer
37 views

Alternative characterization max eigenvalue

I want to see if my line of thinking is correct on the following problem (given that $A$ is Hermitian, the eigenvalues are in non decreasing order $\lambda_\min$ to $\lambda_\max$, and $A$ has at ...
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Are these vectors a base of the Subspace?

$a=(1,2,3,0)$ $b=(0,3,2,4)$ $a$ and $b$ form the subspace $U$ is the base $B(a,b)$ a base of $U$? My guess is yes, because they are linearly indpendent?
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$C^0([a,b])$ is an infinite dimensional vector space

I am proving that $C^0([a,b])$ is an infinite dimensional vector space. The fact that it is a vector space is clear. But I cannot understand how to prove that it has infinite dimension. Let ...
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4answers
90 views

What am I doing wrong here?

Consider this system of equations: $$ \begin{cases} x+y=6\\x-y=5\\2x+3y=7 \end{cases} $$ This is an overdetermined system and doesn't have a solution (the 3 lines don't intersect). But by adding 2nd ...
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2answers
55 views

real matrix with $Tr((A-I)^{T}(A-I) )<1$

$A$ is a $n\times n$ real matrix, $$\operatorname{Tr}((A-I)^{T}(A-I) )<1$$ then $\det(A)\ne0$. well, $$\sum_{i\ne j}a_{ij}^2+\sum (1-a_{ii})^2\lt1$$ How to derived $\det(A)\ne0$? Thank ...
2
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1answer
40 views

Using properties of determinants, show that

Using Properties of determinants, show that: $$ \begin{vmatrix} a & a+b & a+2b\\ a+2b & a & a+b\\ a+b & a+2b & a \end{vmatrix} = 9b^2 (a+b) $$ I've tried it but not ...
0
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1answer
32 views

Find linear transformation of a given matrix of linear transformation

I have two basis. The first is a $R2$ basis ${(1,0) , (0,2)}$. Lets call it basis of $U$. The second is a $R3$ basis ${(1,0,-1), (0,1,2), (1,2,0)}$ Lets call it basis of $V$. Is given a matrix of a ...
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1answer
13 views

Base change Matrix from 2 given Matrices that perform the same function in 2 Bases

I have 2 Matrices. Both of them perform the nilpotent function f, but they are in different Bases. $D_{BB}(U)$ and $D_{CC}(U)$. I need $D_{CB}(U)$. I also have the bases given. How would I go about ...
9
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2answers
104 views

A question about linear subspace and subfield

If we see the complex numbers $\mathbb{C}$ as a linear space on field $\mathbb{Q}$, then $\mathbb{C}$ is infinite dimensional over $\mathbb{Q}$, $2$-dimensional over $\mathbb{R}$, $1$-dimensional over ...
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1answer
74 views

support vector machine - vector algebra

I am trying to understand the basics of SVM algebra, but fail to understand per below: Let us formalize an SVM with algebra. A decision hyperplane can be defined by an intercept term $b$ and a ...