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 ...

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Vector space of $n$-tuples

First off, I would like to do this myself, I'd really like hints on how to proceed so I know where to begin. Let $V_1=\{(a_1, a_2,\ldots , a_n) \mid a_i \in \mathbb C \text{ for } i=1,2,\ldots,n\}$ ...
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1answer
32 views

Addition of two Vector Spaces

If $V$ is the vector space of $n$-dimensional matrices, $U$ is the subspace of lower triangular matrices and $W$ is the subspace of diagonal matrices, would it be correct to say that: \begin{align} ...
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1answer
42 views

Deriving equation in vector notation

I had some trouble deriving an equation from the book 'Elements of statistical Learning' p. 108 equation 4.9. This heavily relies on linear algebra, so I was wondering how the author came to his final ...
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0answers
63 views

Reputation of mathematical journals [on hold]

For a young mathematician what is most convenient for his reputation publishing in "Expositiones Mathematicae" or in "American Mathematical Monthly"?
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2answers
29 views

How many matrices are there in the vector space $M_{m\times n} \mathbb (Z_2)$?

The answer is given to be $2^{mn}$. I know that $\mathbb Z_2$ has 2 elements but I don't get how we're arriving at this solution. Is the number of matrices related to dimension or something?
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4answers
165 views

Proving determinant using properties of determinants

$$\begin{vmatrix} 1 & 1 & 1\\ a & b & c\\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)$$ we have to solve this by using the properties of determinants without ...
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2answers
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An element of $SL(2,\mathbb{R})$

Find the relationship between an elliptic element of $SL(2,\mathbb{R})$ and rotation.. An element $A$ of $SL(2,\mathbb{R})$ is called an elliptic element if $|\text{tr}(A)|<2$ As ...
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1answer
19 views

Showing three lines have a common intersection

Question: Do the three lines $2x+3y=-1$,$6x+5y=0$, and $2x-5y=7$ have a common point of intersection. For this question I made a augmented matrix. My matrix \begin{bmatrix} 2&3&-1\\ ...
2
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1answer
25 views

Finding h so that a system is consistent linear matrix?

How can I determine what value of $h$ would make my system consistent. The augmented matrix of my system is: $\begin{bmatrix} 1&4&-2\\ 3&h&6 \end{bmatrix}$ My first step is to add ...
2
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1answer
39 views

Is a $n\times n$ symmetric matrix necessarely similarly to another symmetric matrix? [on hold]

Given $A\in\mathcal{M}_n(\mathbb K)$ symmetric. Given $H\in GL_n(\mathbb K)$, is $H^{-1}AH$ necessarely symmetric?
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2answers
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Linear Transformation and spanning set

I have the following question below: Let $V$ and $W$ be real vector spaces and $T:V\to W$ be a linear transformation such that $\ker(T) = {0} \subset V$. Let vectors $v_1,v_2,v_3,v_4$ belong to ...
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Concurrency of lines

If the three lines: $$x\sin^2 \theta + y \cos^2 \theta = 1$$ $$x \cos^2 \theta + y \sin^2 \theta = 1$$ $$lx + my + n = 0$$ are concurrent then which of the following is true? a) $l+m=n$ b) ...
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1answer
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Let U;W be vector spaces over R with bases u1; u2; : : : ; um and w1;w [on hold]

Let $U,W$ be vector spaces over $\mathbb R$ with bases $u_1,u_2,\dots,u_m$ and $w_1,w_2,\dots w_n$ respectively. Let $V = U \oplus W$ and linear transformation $P : V \rightarrow U$ be defined by ...
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4answers
44 views

Checking subspace

Let $B$ be a fixed matrix in $\mathbb{R}^{n\times n} $ and $W=\{{A \in \mathbb{R}^{n\times n} :AB=BA}\}$ Then is $W$ a subspace of $\mathbb{R}^{n\times n}$ ? I have tried this so far: a) The zero ...
1
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1answer
33 views

Proof - Inverse of linear function is linear

This is my first proof related to linear functions. It refers to the linear-algebra-$\textit{linear}$ (not the calculus-$\textit{linear}$). Please comment. Theorem The inverse of a linear bijection ...
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2answers
65 views

Quadratic Prime

We had received some questions on Quadratic equations, But I wasnt able to do one. Here it goes: Let $a,b$ be natural numbers $a>1$. Also, $p$ is a prime number. If $ax^2+bx+c=p$ for 2 distinct ...
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0answers
15 views

Logic behind finding determinants and eigen value of a square matrix

To solve a linear equation $Ax = b$ , we need to find the inverse of the matrix $A$. The way by which we find the inverse of $A$ is by dividing a scalar by adjoint of the matrix $A$. We call this ...
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3answers
112 views

Proving determinants using properties of determinants

$$\begin{vmatrix} 1 & a^2+bc & a^3\\ 1 & b^2+ca & b^3\\ 1 & c^2+ab & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a^2+b^2+c^2)$$ we have to solve this by using the properties of ...
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2answers
22 views

Linear Transformations and independence

I have the following question: Let $V$ and $W$ be real vector spaces and $T:V\to W$ be a linear transformation such that $\ker(T) = {0} \subset V$. Let vectors $v_1,v_2,v_3,v_4$ belong to $V$, ...
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0answers
42 views

What shall the characteristic of K be for the identity (1) to be valid?

Let $K$ be a field. Let $f$ be a linear operator on $V$. Consider the identity: $v=1/2(v+f(v))+1/2(v-f(v))$ (1) Suppose, $f$ is such that for any $v$ in $V$, one has: $f(f(v))=v$ What shall the ...
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1answer
38 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
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2answers
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Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
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Alternative proof: Matrix $A$ is similar to $B$ iff $\lambda I - A$ is equivalent to $\lambda I - B$

We have this theorem for square matrices: If $\lambda I - A$ is equivalent to $\lambda I - B$, then $A$ is similar to $B$. ($A$, $B$ are matrices in $K^{n\times n}$, $K$ is a number field, ...
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2answers
418 views

proof that a continuous additive homomorphism $\mathbb{R}^n\to\mathbb{R}^m$ is $\mathbb{R}$-linear

How can we prove that a continuous additive homomorphism $ \Phi \colon \mathbb{R}^{n}\to \mathbb{R}^{m} $ is $\mathbb{R}$-linear. i.e. satisfies $ \Phi (rv)=r \Phi (v)$ for $ r\in \mathbb{R} $ and $ v ...
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2answers
56 views

How to determine whether it is vector space? [on hold]

Does the set of all polynomials of degree exactly $5$, together with all the constant polynomials,determine a vector space?
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1answer
18 views

Can the coefficients used to prove a set of functions is linearly dependent be imaginary?

Example: $\cos x$, $e^{ix}$, $3\sin x$. I can show: $C_1\cos x + C_2 e^{ix} + C_33\sin x = 0$ if $(C_1,C_2,C_3) = (1,-1,i/3)$ But i don't know if $C_3 = i/3$ is a valid coefficient to choose. Can ...
5
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2answers
106 views

An interesting linear algebra question

Let $A$ and $u$ be $n\times n$ matrix and $n\times 1$ vector of $\mathbb{C}$. Denote $\overline{A}$ is the matrix $(\overline{A})_{ij}=A_{ij}^*$, the conjugate number; ($\overline{A}$ is not the ...
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2answers
343 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
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0answers
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What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
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1answer
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Linear Transformation $T-T^2=I$

Let T be a linear transformation from a vector space V over reals into V such that $T-T^2=I$. Show that T is invertible Solution: I started by multiplying T on both sides and getting $-T^3=I$
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0answers
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How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
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1answer
21 views

Question about proof of Cauchy-Schwarz inequality.

I was trying to prove the Cauchy-Schwarz inequality, and came up with the following: $$ |u||v|\cos{\theta} \leq \frac{1}{2}|u|^2 + \frac{1}{2}|v|^2 $$ I got stuck here, did some googling and found a ...
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0answers
16 views

Generalized Schur complement theorem

Let $M$ be an $(n+m)\times(n+m)$ real non-symmetric positive semidefinite (PSD) matrix partitioned as \begin{eqnarray*} M=\left(% \begin{array}{cc} A~~B\\ C~~D\\ \end{array}% \right), ...
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0answers
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Curves through points [on hold]

An astronomer wants to determine the orbit of an asteroid about the sun. He sets up a Cartesian coordinate system in the plane of the orbit with the sun at the origin. By Kepler's law the orbit must ...
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1answer
20 views

The nullity of a square matrix with linearly dependent rows is at least one. TRUE OR FALSE

Here is the answer my textbook gives. http://imgur.com/ycCRoWK I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
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1answer
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TRUE OR FALSE: Matrices with linearly independent row and column vectors are square.

Here is the answer of my textbook: http://imgur.com/vEoY31O Why must a matrice with linearly independent vectors have nullity(A)=0? That is where I lose track of the question. Are zero rows ...
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1answer
42 views

Find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace spanned by $g(x)=x−12$ and $h(x)=1$.

Use the inner product $\langle f,g\rangle =\int_0^1 f(x)g(x)dx$ in the vector space $C^0[0,1]$ to find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace $V$ spanned by $g(x)=x−1/2$ and ...
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3answers
77 views

$A^2=cA$ for some $c \neq 0$

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $$A^2 = cA.$$ Questions. 1) Is there a matrix class for matrices ...
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0answers
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Modifying U=mxn SVD Algorithm to U=mxm Algorithm

I have painstakingly ported this Python source "svd.py" to C++. I confirm it works for the example it comes with. While testing another example (this one, from Wikipedia), the assert statement trips ...
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1answer
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what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
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1answer
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Normal Matrix Having all real eigen values is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
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1answer
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On Stochastic Matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
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+50

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
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1answer
96 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
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3answers
62 views

Linear Algebra problem (related to transpose matrices)

Producing $x_1$ trucks and $x_2$ planes requires $x_1+50x_2$ tons of steel, $40x_1+1000x_2$ pounds of rubber, and $2x_1+50x_2$ months of labor. If the unit costs $y_1, y_2, y_3$ are \$700 per ton, \$3 ...
1
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1answer
48 views

A solution of a linear system in some extension field implies a solution in the subfield

Fix a field extension $k\subseteq K$ and consider a linear system $Ax=b$ where $A$ is a matrix (not necessarily square) with coefficients in $k$. I don't understand why if the above linear system ...
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1answer
25 views

Prove that a normal matrix is unitary/Hermitian

I'm stuck with these two questions for while. I'd appreciate your help. ...
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2answers
42 views

How do you calculate this third eigenvector in this 3x3 matrix?

Scroll down to the bottom if you don't want to read how I arrived at my original two answers. My question is how are all the online calculators I check coming up with this third eigenvector (1, 1, ...
4
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3answers
134 views

Spectrum of matrix with single scaled row

Let $M$ be a real symmetric positive-definite matrix and $D_a$ the diagonal matrix $$D_a = \left[\begin{array}{ccccc}a & & & &\\& 1 & & &\\& & 1 & ...
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2answers
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Problem book for abstract linear algebra

Kindly suggest a good book for abstract linear algebra other than finite dimensional vector space by P R Halmos