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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Assuming a ray defined by a starting point and a direction. How can I tell if a plane is behind it or in front of it?

If I have a ray defined by a starting point and a direction, and a plane defined by its normal and its distance from the origin, how can I tell if the plane is in front versus behind the ray? By ...
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27 views

3 x 3 linear system organization [duplicate]

How to organize this 3x3 linear system in order to solve it with determinants afterwards.
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44 views

Finding maximum no of $1$'s

We are given a matrix $A \in M_n (F)$ such that all its entires are either $1$ or $0$. I need to find the maximum number of $1$'s that can be in matrix $A$ so that it is still invertible. My try : ...
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61 views

Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
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$3\times3$ linear system organization

How to organize the system below? Especially the 2nd row of the system. $$\left\{\begin{eqnarray} 4x-3y+2z+4&=&0\\ x-\frac y3+\frac z2&=&-\frac16\\ 5x+2z&=&3y-3\\ ...
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138 views

How prove this matrix $B^{-1}-A^{-1}$ is positive-semidefinite matrix,if $A-B$ is positive matrix

Question: Let $A,B$ be positive $n\times n$ matrices, and assume that $A-B$ is also a positive definite matrix. Show that $$B^{-1}-A^{-1}$$ is a positive definite matrix too. My idea: ...
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58 views

Representing a Linear Map in a new basis

Consider the linear map $\mathcal{A}:\begin{bmatrix}x \\ y \end{bmatrix} \to \begin{bmatrix}-5x+9y \\ -4x+7y \end{bmatrix}$ Obviously this is represented by the matrix $A=\begin{bmatrix}-5 & 9 \\ ...
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36 views

Making a matrix invertible

Given $N$ distinct real numbers $x_1,\ldots, x_N$, how can I show that there exist real numbers $a_1,\ldots, a_N$ so that the following matrix is invertible? $$\begin{bmatrix} \exp(ia_1 x_1) & ...
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Dimension of vector space of 2x2 skew symmetric matrices

I had a question about the dimension of this subspace. This was related to a problem that had a case of n x n matrices, but I accidentally read it as the special case of 2x2, but never the less the ...
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1answer
78 views

expansion of matrix inverse

I would like to invert a square matrix $L$. One can write it as a sum of two matrices, one containing the diagonal terms ($D$) and the other the off-diagonal ones ($A$). $$L = D+A$$ I would like ...
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22 views

slanting of non-square matrix

I'm looking for an operation to ''slant'' a (not necessarily square) matrix. I want this: $ \begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l ...
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24 views

Assigning a variable as constant and find out the minimum value

I have got a problem in assigning a variable as constant and solving a equations for minimum value. These are my equations: kd (2 h2 Cos[[Theta]2] - b2 Sin[[Theta]2]) (b1 - (b1 + b2) ...
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48 views

Finding counterexamples and proving a transformation is linear

Can someone please explain instances where $ f^{-1} (f (A)) \not = A $ and $f (f^{-1}(B)) \not = B$ if $ f:X \rightarrow Y $ and $ A$is a subset of X and B is a subset of Y? I can't think of when this ...
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2answers
28 views

Linear system with parameters

How do I find a condition on the parameters a,b,c so that this system will have a solution? $x_1 + x_2 + x_3 = a$ $x_1 + 6x_2 + 3x_2 = b$ $3x_1 -2x_2 + x_3 = c$
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105 views

Why does the discrete cosine transform compact the information at the “low frequencies”?

I've been investigating about the discrete cosine transform. I think I understand the practical applications it has and how it is used in image/audio compression. I also know it is related with the ...
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64 views

General Linear Systems

Determine if the following systems are compatible and, if so, find the general solution: $2x_1-6x_2+4x_3=2$ and $-x_1+3x_2-2x_3=-1$ How can I set this system into a matrix to determine if they are ...
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133 views

Geometry question pertaining to $4$ points in the plane where $90$ degree projectors are on each point and we must illuminate the whole plane.

Suppose we have $4$ points that can be positioned anywhere in $\mathbb{R}^2$. Now imagine each point has $90$ degree projectors coming out of them and you can rotate these projectors any way you would ...
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92 views

Hamel Basis Exercise Proof Clarification.

While looking up something else on stack exchange, I ran across this question An exercise about a Hamel basis and it intrigued me. The answer was provided by Jonathan Golan ...
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1answer
95 views

Optimization: KKT conditions statement

I'm currently following this material Optimization Theory: Chapter 2 Theory of Constrained Optimization And I can't understand why the following statement is true, between the equations (2.9) and ...
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2answers
66 views

Prove that S is diagonal

Let $S: V\rightarrow\ V$ be an operator on an $n$-dimensional real vector space with an eigenvalue that has geometric multiplicity equal to $n-1$. Prove that $S$ is diagonal. Give an example of such ...
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Solve the algebra equation- unsure about order of operations, how to go about solving, solve for x

The question states: solve the equation. State the solution set and check your answer. I've spent a good 45 minutes on this, to know avail. If someone could sort of walk me through this I would be ...
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1answer
32 views

Finding values for which a bilinear form is an inner product

I am trying to find the values (if any) of p and q for which the following satisfies the definition of an inner product: $$ \left \langle \mathbf{z}, \mathbf{w} \right \rangle = z_1\overline{w_1} + ...
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43 views

Stuck on condition number derivation of the perturbed equation $(A + \Delta)\tilde{x} = b + \delta_b$

I've almost got what I want. We start with $Ax = b $ and $(A + \Delta)\tilde{x} = b + \delta_b$. What I have then is \begin{align*} \tilde{x} - x &= -A^{-1}\Delta\tilde{x} + A^{-1}\delta_b \\ ...
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65 views

Why a matrix is a linear map

I know $L:\mathbb{R}^n \rightarrow \mathbb{R}^m $ by the formula $L(x)=Ax$ is a linear map. But I cannot understand why the matrix $A$, just itself, is a linear map. This question came to my mind by ...
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22 views

Linear algebra (Coordinates)

Question: Find the coordinates of $x=(1,0,0)$ in relation to base $$B=\{(1,1,1),(-1,1,0),(1,0,-1)\}.$$ I tried: $a,b,c\in R$ such that $$a(1,1,1)+b(-1,1,0)+c(1,0,-1)=(1,0,0)=x$$ but I'm not sure ...
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81 views

Correctly negating “there exists a subset of $S$ that is a basis for $V$”

I would like to prove the following by contradiction: "Let $V$ be a vector space having dimension $n$, and let $S$ be a subset of $V$ that generates $V$. Prove that there is a subset of $S$ that is ...
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3answers
5k views

Find all matrices that commute with given matrix

Find all $2\times 2$ matrices that commute with $$\left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right)$$ My progress: I know that a square matrix commutes with itself, the identity ...
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3answers
46 views

Find matrices A and B

I am coming across questions like these: Find $A$ and $B$ if $$2A+3B=I_2$$ $$ A+B=2A^t$$ $$2A+3B= \left( \begin{array}{cc} 8 & 3 \\ 7 & 6 \end{array} \right)$$ $$A+B^t= \left( ...
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69 views

Fourier Interpolation

I have this Equation, that I modeled from my measurements and simulations. $I^{exp}_{l,m} = (\mathbf{F}^{H}.\mathbf{A}.I^{true})_{l,m}$; $H$ is the Hermitian transpose and $\mathbf{F}^{H}$ is a block ...
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45 views

dimension of a subspace

How do we find out the dimension of the set of possible values of a vector from a set of linear equations in general? For example, the set of values taken by vector $x=[x_1, \dots, x_n] \in R^n$ that ...
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44 views

Prove that the set of all fixed points is a hyperplane

Let $A$ be a $n$-dimensional affine space ($ 2 \leq n$) and let $\Phi:A\to A$ be a bijective affine mapping, which isn't the identity mapping, with the following property: For all points $p$ and $q$ ...
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76 views

Cartesian Equation and Parametric Equation Help

I just need some help with a maths question that I am trying to get done for a maths tutorial homework sheet. The question is... Let L be the line through D = (6,5,4) and E = (1,0,6), and let P be ...
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Direct limits of free abelian groups and diagonalization

So, say I have a matrix $A\in M_d(\mathbb{Z})$ and would like to describe the group $\lim(\mathbb{Z}^d,A)$, i.e. the limit of the stationary system $$ \mathbb{Z}^d\to^A \mathbb{Z}^d \to^A ...
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62 views

Bounds on the singular values of a matrix with unitary columns

if $X$ is a matrix with unitary columns ( each column has unit norm ), are there lower and upper bounds on the minimum and maximum singular values of $X$? I could prove a lower bound for ...
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136 views

compute the bisecting normal hyperplane between two $n$-dimensional points.

I have two points $\mathbf{x_1}$ and $\mathbf{x_2}$, where $\mathbf{x_i}=\{x^i_1, x^i_2, \ldots, x^i_n\}$. I need to find a normal hyperplane $P$ that goes through the midpoint of $\mathbf{x_1}$ and ...
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120 views

Efficient algorithm to find a minimum spanning set for a given vector.

A few days ago a colleague proposed the following problem. Let $W$ be a finite subset of a vector space $V$, and let $v\in\langle W\rangle$ (the linear span of $W$). Is there an efficient ...
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Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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838 views

Find values of h such that the vectors (2, 4) and (h, 6) span $\mathbb{R}^2$

My homework is asking me to answer problems such as the one that follows: Find all values of $h$ such that the vectors $\{a_1, a_2\}$ span $\mathbb{R}^2$, where $a_1 = (2, 4)$ and $a_2 = (h, 6)$. I ...
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292 views

Is this Determinant and Trace identity equivalent to Unitary matrix?

Thanks for any help in advance. I have this equality for a 2x2 invertible complex matrix: $$\text{Tr}(AA^*)=2|\text{det}(A)|^2$$ where $*$ is complex conjugate transposition. Is this equality ...
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28 views

Linear Independence with distinct variables

If there is a group of vectors $v$ such that $v=\left(\begin{array}{c} 1\\1 \end{array}\right), \left(\begin{array}{c} x_1\\x_2 \end{array}\right), \left(\begin{array}{c} x_1^2\\x_2^2 ...
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217 views

Least squares fitting using cosine function?

Hello I am trying to fit a harmonic of the form $$y = b + c\cos(x)$$ to four data points (0,6.1) (.5,5.4) (1,3.9) (1.5,1.6) using least squares for homework. I know that the error $= Y_i - f(x_i)$ but ...
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Positive Semi Definite Matrix

If $A$ is a positive semi definite matrix, is $\left[ \begin{matrix}c_1A & c_2A \\ c_3A & c_4A\end{matrix} \right]$ positive semi definite? ($c_1, c_2, c_3, c_4 > 0)$ In general, what ...
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Polar correlation and conics in $\Bbb RP^2$

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
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Vector space without a scalar product

In linear algebra the terms vector space and scalar product always (at least for me) appear together. Can you give me an example of a vector space without a scalar product? Does the senescence Let V ...
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54 views

$\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ . what can you say about the direction of $\vec{b} \times \vec{c}$?

I know that $\vec{a} \times \vec{b}$ and $\vec{c} \times \vec{d}$ are perpendicular therefore the dot product would equal $0$.
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$T:E\longrightarrow F $ isomorphism. Prove $\dim \langle \vec x_1,\ldots ,\vec x_n\rangle =\dim \langle T(\vec x_1), \ldots,T( \vec x_n)\rangle$.

Let $T:E\longrightarrow F $ be an isomorphism. Prove $\dim \big\langle \vec x_1,\ldots ,\vec x_n\big\rangle =\dim \big\langle T(\vec x_1), \ldots,T( \vec x_n)\big\rangle$, for every $\{\vec x_1, ...
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37 views

Linear system and subspaces

Let $S $ be a subspace of $R^n$ with dimension k and $m = n-k.$ Show that $$\exists A \in R^{m\times n}, b\in R^m$$ Such that $$S = \{ x \in R^n : Ax = b\}$$ My attempt consist of getting m ...
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47 views

Find base and dimension of given subspace

Let $T$ $\in M_{4}(\mathbb R)$ and consider $S= \{M \in M_{4\times1}|T.M = 0\}$. In the case ...
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23 views

How to know which variables to parametize in a large matrix?

(dont want anyone to solve the problem, just don't understand one thing) So I have a homework problem where I got a 3x6 matrix, and I have to parametrize the equations and solve for each variable in ...
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74 views

Solving system of linear equations in the field $Z_p.$ [duplicate]

Consider $5x+3y=4$ and $3x+6y=1.$ List the set of primes for which this system of linear equations does not have a solution in the field $Z_p.$