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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Solving linear equation with three variables

1)Do equations with 3 variables require at least 3 equation for us to solve without any dependent variable? 2)if two equations of such kind equate to zero for eg: a-b+2c=0 and 3a+b+c=0. Then using ...
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82 views

$V = W\oplus W^\perp$

Let $W$ be a vector subspace of a finite dimensional vector space $V$ over $\mathbb{C}$ and let $k:V\times V\to \mathbb{C}$ be a nondegenerate bilinear form such that $k_{|W\times W}$ is also ...
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645 views

In a survey of 270 college students, it is found that 64 like brussels sprouts

In a survey of 270 college students, it is found that 64 like brussels sprouts, 94 like broccoli, 58 like cauliflower, 26 like both brussels sprouts and broccoli, 28 like both brus- sels sprouts and ...
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245 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
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291 views

Way to determine eigenvalues from 3. degree characteristic equation?

I have derived the following characteristic equation for a matrix $$a^3 - 3a^2 - a + 3 = 0$$ where $a = \lambda$. I know that it's possible to find the roots (eigenvalues) by factorization, but I ...
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126 views

correlation matrix of the random matrix

$\mathbf{X}$ is random matrix given by $\mathbf{X}=\left[\begin{array}{*{20}{c}} \mathbf{x}_{1}&{}&{}&{}\\ {}&\mathbf{x}_{2}&{}&{}\\ {}&{}& \ddots &{}\\ ...
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Relation between determinant and matrix rank

Let $A$ a square matrix with the size of $n \times n$. I know that if the rank of the matrix is $\lt$ $n$, then there must be a "zeroes-line", therefore $det(A) = 0$. What about $rank(A)=n$? Why ...
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113 views

Proof involving a summation

How would I go about proving that $\sum_{i<j} 1 $ = $ n\choose 2$ and $$ \sum_{i<j}(x_i +x_j) = (n-1)\sum x_i $$ I understand the intuition behind the statements. I'm just unsure of how to ...
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63 views

Projection of $V$ onto $U$ along $T$ ?

I need some help with my proofs. :) Let $ V = U_1 \oplus W = U_2 \oplus W $ and $\pi_1 : V \to V$ the projection of $V$ onto $U_1$ along $W$ and $\pi_2: V \to V$ the projection of V onto $U_2$ along ...
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40 views

Linear transformations of the real polynomial space

Let n a natural fixed number and X the space of all real polynomials of degree at most n. I need to give a basis for X and say what of these following transformations are linear in X in X, this is ...
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37 views

Similarity transform with psuedoinverse

If $P$ is of full rank and $A$ and $P$ are square, $PAP^{-1}$ is a similarity transform of $A$. Notably it will have the same eigenvalues as $A$. Is there anything useful we can say if $P$ doesn't ...
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147 views

$M$ and $N$ are matrices that satisfy $MNMN=0$,what can we say about $NMNM$. [duplicate]

From my personal point of view, we cannot deduce that $NMNM=0$,but I can't find a counterexample.
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84 views

What's this called, and how do you do it mathematically?

So, I'm teaching myself physics, and I have limited knowledge of calculus(I'm taking my first high-school class next semester in math). I'm attempting to calculate torque, but this is math, not ...
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43 views

Dimensionality of a subspace described by equations

T-F questions 20-21 from http://www.math.washington.edu/~smith/Teaching/308/2013-Spring-Midterm-Answers.pdf The set $\{(x_1, x_2, x_3, x_4) \mid x_1 + x_3 = x_2 - x_4 = 0\}$ is a two-dimensional ...
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233 views

Question describing the span of 3 vectors using a single equation of a plane

T-F: The linear span of the vectors $(4,0,0,1)$, $(0,2,0,-1)$, and $(4,3,2,1)$ is the 3-plane $x_1 - 2x_2 + 3x_3 -4x_4 = 0$ in $\mathbb{R}^4$. I can substitute each vector into the equation and ...
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117 views

Dual spaces, Find a natural isomorphism between V and $(V^{*})^{*}$

Let me just start by saying I'm very very new to this material. I have very little idea what's going on. I've red wikipedia and a few other sources but this is still very hard for me, so I would ...
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653 views

A Householder matrix is symmetric

I want to show that a Householder matrix is symmetric, so I must show that $H^T = H$, but from the formula $$H= I - (uu^T/\beta),$$ they are not equal. What's wrong with my reasoning? EDIT: I ...
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57 views

Show a matrix is normal - check my proof

Short easy question, I just want someone to double check what I did. We are given that $T$ is an invertible, normal matrix. We are asked to show that $T^{-1}$ is also normal, and find it's unitary ...
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68 views

$A$ be a $n\times n$ real matrix, its diagonalizability with specific conditions

$A$ be a $n\times n$ real matrix, could any one tell me which of the following is correct? If $A^2=0$, then $A$ is diagonalizable over $\mathbb{C}$. If $A^2=I$, then $A$ is diagnalizble over reals. ...
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101 views

Hermitian matrix such that $4M^5+2M^3+M=7I_n$

$n$ is a positive integer. Besides the identity matrix $I_n$, does there exist other $n\times n$ Hermitian matrix $M$, such that the following equality $$4M^5+2M^3+M=7I_n $$ hold? I try this: ...
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134 views

Orthographic projection matrix

I'm trying to find the matrix of a generic orthographic projection onto a given plane $A$ that passes at the origin and the n-vector as the normal vector to the plane. I only found matrices related ...
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52 views

Solutions for linear equation system Ax = b

Two solutions are known for the linear equation system $Ax = b$. These are (for example): $$ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} $$ How to determine (at least) ...
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80 views

Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
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34 views

Why is this presupposition necessary? (set)

Let $U_i$ be subspaces of a vector-space V. If $ U_1 \subseteq U_3 $ then follows: $U_1 + (U_2 \cap U_3) = (U_1 + U_2) \cap U_3$. I have already proven this equality. I nood have to show why the ...
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853 views

How to determine vector space?

I am taking a linear algebra course, and we are currently learning about vector spaces and subspaces. On the beginning of the chapter it is said that vector space must "comply" with all of the ten ...
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145 views

$ABAB=0_{n\times n}$, then must $BABA$ be $0_{n\times n}$?

$A, B$ are $n\times n$ Matrix, and $ABAB=0_{n\times n}$. Could we conclude that $BABA$ must be $0_{n\times n}$? I fail to give a counterexample when $n=2$, so I guess the answer is "yes"
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597 views

Comparing two covariance matrices

In the textbook I am reading they use positive definiteness (semi-positive definiteness) to compare two covariance matrices. The idea being that is A-B is pd then B is smaller than A. But I'm ...
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30 views

Subspaces of linear combinations

What are sub-spaces of linear combinations? How to tell the basis? I'm not sure what it is. Might this refer to linear span?
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320 views

How to construct change of basis matrix

How do I construct a change of basis matrix? For example in $\mathbb R^3$, how to construct matrix changing basis from $A$ to $B$? $A=\begin{pmatrix} 1 \\ 0 \\5 \end{pmatrix}\begin{pmatrix} 4 \\ 5 ...
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223 views

necessary condition for subspace of a vector space

Currently I'm reading linear algebra books of leon's and friedberg's. In friedberg's book, for being subspace, a subset of vector space should (1). contain zero vector (2). closed under scalar ...
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129 views

Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal.

The question is: Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal. Then I have to find the spectral decomposition of $T^{-1}$. At first I tried to prove it by ...
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43 views

linear equation equality constraint - which row to remove?

In this question, there is a linear equation $Ax=b$, and a particular element of $x$ is constrained to a known value. The "move to the right hand side" approach (see below) results in an ...
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1answer
51 views

Given 9 positive numbers taking $N$ distinct values, what is minimum $N$ so that they can be arranged into an invertible square matrix?

Assume that $(a_1, \dotsc, a_9)$ are different positive numbers. Let us make a $3\times 3$ matrix $A_S$ by putting them arbitrarily into 9 positions available. Show that there is always ...
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79 views

Looping over $k$-element subsets by switching elements

I would like to iterate over the $k$-element subsets of $\{1,2, \dots, n\}$ in a natural way by switching elements. Two subsets $v,w$ are adjacent if $|v \cap w| = k-1$ or equivalently if their ...
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228 views

Can I define a plane given 2 points in xyz coordinates as well as roll angle about that vector?

I am working on a complex motion analysis, trying to calculate wrist angles in 3 dimensions. I have sensors placed as this diagram depicts and need both flexion/extension angles as well as ...
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122 views

Must vectors in $\mathbb{R}^n$ have their “tail” at origin?

I was looking the definition for an $n$-sphere centered at origin with radius $r$: $$\mathbb{S}^n = \{v \in \mathbb{R}^{n+1} : ||v|| = r \}$$ Although I understand that the $||v|| = r$ condition ...
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151 views

Coordinate Transformations

I am physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) between coordinate transformation of two kinds : Rotation of coordinate (and ...
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4answers
123 views

Can this whiskey be a result of combining whiskey A, B, and C in any ratio?

There is a whiskey made up of 64% corn, 32% rye, and 4% barley that was made by blending other whiskies together. I am trying to figure out if there is a chance the ratio of this whiskey could be the ...
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83 views

study of subspace generated by $f_k(x)=f(x+k)$ with f continuous, bounded..

Let $f:ℝ→ℝ$ be continuous, bounded function such that the space $\mathrm{lin}\{f_k(x)=f(x+k)∣k ∈\mathbb{N}\}$ is finite-dimensional. Determine an expression of f. I started with: Let $P_f(X) = ...
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1answer
55 views

complex irreps is in bijective correspondence with sequences

Let $\{a_n\}$ be a sequences of positive integers such that $$0 \leq a_n\leq p^n - 1,$$ $$a_n \equiv a_{n +1} \bmod p^n \quad \text{for all $n$}$$ Prove that the complex irreps of the group $ ...
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30 views

Projective homologies

In a projective plane (i.e. two-dimensional) $\mathbb P$, we call a general homology a projective transformation $h:\mathbb P\to\mathbb P$ such that $h$ has a line of fixed points $L$ called the axis ...
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97 views

Square matrix and determinant inequality

Let $A, B, C$ be invertible $n \times n$ square matrices with $AC=CA$ and $B^2C^2=I_n$ Is $\det(ABC +CBA +A^2+I_n)$ always $\geq 0$?
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100 views

A canonical form for this equivalence relation on matrices

This question is inspired by http://cs.stackexchange.com/q/19250/755. Define the equivalence relation $\sim$ as follows: If $M,N$ are two $8\times 8$ (or $n\times n$ if you prefer generality) $(0,1)$ ...
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64 views

Why is this vector-set linear-dependent?

$$\{ (1,0),(0,1),(0,0)\} $$ Maybe I'm losing it, but I can't see here a vector which is a linear-combination of the other two.
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106 views

Are $A,B$ similar matrices? Check my proof

We are given $A,B$ are orthogonal $4$ by $4$ matrices with real values only. We are given $\det(A) = \det(B) = 1$ and $\mathrm{trace}(A) = \mathrm{trace}(B)$. Is $A$ similar to $B$? My solution: I ...
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57 views

if $Ax=b$ has no solution $\Rightarrow$ $b$ is a linear combination of $A$'s columns.

if $Ax=b$ has no solution $\Rightarrow$ $b$ is a linear combination of $A$'s columns. I know the statement is false but, please help me understand why. Thanks
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38 views

Dimension of a vector space when sum and multiplication changes

If a vector space over the complex numbers has dimension $n$, can we change the definitions of sum and multiplication by complex numbers so that the dimension changes?
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81 views

$\alpha^{-1}(\ker(\beta))$, how to find? [closed]

I can't understand how to find $$ \alpha^{-1} (\ker(\beta)) $$ where: $$ \alpha = \pmatrix{1 & 2 & 1\\0 & 1 & 0}\\ \beta = \pmatrix{0 & 1\\ 0 & 1 } $$
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54 views

Problem with indefinite square roots

I have a question which reads: If $$\sqrt{12 + \sqrt{12 + \sqrt{12 + \cdots\cdots}}} = x$$ Then the value of $x$ is _. I think that we can write $$x^2 - 12 = \sqrt{12 + \sqrt{12 + \sqrt{12 + ...
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274 views

vector projection on a unit ball

On an article I'm reading, I find that: if $v$ is a vector, the projection of of $v$ on the unit ball is: $$p(v)=\frac{v}{\max\{1,\|v\|\}}$$ I know that a projection of a point $v$ into a space is the ...