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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Linear algebra: orthonormal eigenvectors

Let $A=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ Its eigenvalues are $\lambda=0,0,3$. Find two pairs of orthonormal eigenvectors for $\lambda=0$ and ...
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What is the necessary condition for a matrix to have eigenvalue 1?

What would be the necessary condition for a matrix of any $n \times n$ to have eigenvalue 1? I know that it must have a corresponding eigenvector - that is obvious - I want to know things like how ...
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133 views

Linear Algebra Proof

If A is a $m\times n$ matrix and $M = (A \mid b)$ the augmented matrix for the linear system $Ax = b$. Show that either $(i) \operatorname{rank}A = \operatorname{rank}M$, or $(ii)$ ...
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calculating points in specific distance from given point & direction

I have a 3-dimensional view, where I have drawn a line $L$. I know the line direction vector $(x,y,z)$, where $L$ is also the center of a cylinder with given radius $r$. I wish, based on the radius ...
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5 linear equations in 5 unknowns

I need an example of 5 linearly independent equations with 5 variables. How can I write such a equation set. As an example: ...
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161 views

System of equations is valid, but cannot solve, since I find a square root of negative number

I was solving a problem that required me to find some points in the plane perpendicular to a segment that I already had. Since I am applying the fact that the dot product of the vector I already have ...
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Linear transformations and constant vectors

Let's call a column vector in $R^n$ constant if all the entries of it are the same. Let $A:R^n\rightarrow R^n$ be a linear transformation and let $B$ and $C$ be two bases in $R^n$. Suppose that $b_1 + ...
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35 views

Problem involving rank

Let's call a sequence semi-arithmetic if the difference of any two consecutive elements of it can take two different values only.(E.g the sequence $3,7,11,12,16,17,18$ is semi-arithmetic). For the ...
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Help with Linear Transformations

Working through a homework problem from my linear algebra course. We're using the Gareth Williams Linear Algebra with Application - 7th Edition. The question comes from section 5.2: Matrix ...
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Symmetric matrix is always diagonalizable?

I'm reading my linear algebra textbook and there are two sentences that make me confused. (1) Symmetric matrix $A$ can be factored into $A=Q\lambda Q^{T}$ where $Q$ is orthogonal matrix : ...
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Prove concurrency of triangle altitudes with vector algebra?

I know how to do it in normal Euclid geometry, but is it possible to do it with vector algebra?
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562 views

Calculating the dimension of a vector space in 2 different ways

Let $T$ be the linear transformation represented by the matrix $$ \left( \begin{array}{ccc} 1 & 1 & 0 & 3 \\ 1 & 1 & 1 & 5 \\ 2 & 2 & 1 & 8 \end{array} \right) $$ ...
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Reconstructing a Matrix in the $\Bbb{R}^3$ space with $3$ eigenvalues, from $3$ matrices in $\Bbb{R}^2$ or $\Bbb{ C}^2$ space with $2$ eigenvalues

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
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158 views

prove 2 ideal has the same monic generator

Can anyone help me with this problem. I try many ways to solve this but still can't find the solution.Thanks Let $K$ be a subfield of field $F$, and suppose $f$, $g$ are polynomials in $K[x]$. Let ...
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55 views

How to correct this diagonalization argument?

I want to diagonalize (with a orthogonal change of coordinates) the quadratic form $F(x,y)=x^2-\frac{n-2}{\sqrt{n-1}}xy-y^2$. I already know that the system $(*)$ bellow \begin{eqnarray} ...
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Procedures to find solution to $a_1x_1+\cdots+a_nx_n = 0$

Suppose that $x_1, \dots,x_n$ are given as an input. Then we want to find $a_1,\ldots,a_n$ that satisfy $a_1x_1 + a_2x_2+a_3x_3 + a_4x_4+\cdots +a_nx_n =0$. (including the case where such $a$ set does ...
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248 views

Determinant of anti-diagonal permutation matrix

Consider a $5\times5$ matrix $P=(5,4,3,2,1)$ which means it has anti-diagonal entries of $1$'s. If we calculate $\det P$ using the theorem "The determinants changes sign when two rows are ...
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Fibonacci sequence for the determinants

It's really easy question but I think I'm missing some points. Let $F_n$ is $n$ by $n$ matrix $$F_n=\det \begin{pmatrix} 1 & -1 & & & & \\ 1 & 1 & -1 & & & ...
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413 views

Dimensions of vector spaces in an exact sequence

I've read the following formula in wikipedia: Given finite dimensional vector spaces $V_i$ and an exact sequence $\cdots\rightarrow V_i\rightarrow V_{i+1}\rightarrow\cdots$, we have $$ \sum_{n\in ...
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Linear Algebra (linear transformations)

13. Suppose $V$ and $W$ are finite-dimensional vector spaces and $T:V \to W$ is an isomorphism. Then there exist bases $\mathcal{B}$ and $\mathcal{C}$, for $V$ and $W$ respectively, such that ...
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175 views

To check whether $\mathrm{rank}(A)$ = $\mathrm{trace} (A)$

Let $A \in \mathrm{Mat}_{n\times n}(\mathbb C)$ with $A^2 = A$. Does it always imply that $\mathrm{rank}(A)$ = $\mathrm{trace}(A)$?
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Probability of a matrix having determinant zero

What would be the probability of matrix having determinant zero out of all matrices with all entries being positive? How does one calculate such? Edit: Restriction to natural numbers and size of $n ...
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Solving system of linear differential equations using diagonal matrix

This is my first time solving a problem like this and I just wanted to make sure if what I did was correct. \begin{equation} x'_1=5x_1 + 2 x_2 - x_3 \\ x'_2=-2x_1 + x_2 - 2x_3 \\ x'_3=-6x_1 - 6 x_2 ...
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Eigenvectors of similar matrices

If $A$ and $B$ are similar matrices then every eigenvector of $A$ is an eigenvector of $B$. Is the above statement is true? I know that similar matrices have same eigenvalue but I'm not sure about ...
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Are these two eigenvectors equivalent? (Easy Question)

I solved this homework question and when I compared my solution to the textbook solution, the eigenvector is slightly different. I got $[0,\frac{1}{2},1]$ The book says: $[0,1,2]$ It's basically ...
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Determining if a point lies on a particular slope, between two points

This might be a simple question, but I'll ask anyways. I've been reading up on the basics of calculating the slope using two points: m = y1-y2/x1-x2 Which after that, I can figure out the equation of ...
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64 views

Uniqueness of inverse matrix and possibility of $P=PX$

Suppose that there is non-zero vector $P$ of size $1 \times n$. 1) Does there exist some $P$ that $P=PX$ without $X$ being identity matrix? 2) When $AB = BA = I$ and $A$ given, can there be several ...
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382 views

Why is a singular matrix rare?

I am exploring patterns of integers in $n\times n$ matrices. I have two matrices that have a determinant of $0$ and a circulant matrix that has positive determinants that differ depending on $n$. I ...
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50 views

eigenvalue of P9 of the differential operator

let $V$ be the vector space consisting of polynomial with real coefficients in the variable $t$ of degree $≤ 9$.let $D:V→V$ be the linear operator by $D(f)=df/dt$.then $0$ is an eigenvalue of D. is ...
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302 views

Finding a non-diagonal matrix that can operate similar to a diagonal matrix

Suppose we have diagonal square($n \times n$) matrix $A$. (entries on diagonal are non-zero.) 1) Can there be non-diagonal square matrix $C$ that there exists non-zero vector $B$ with all entries ...
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Solve for eigenvector (complex eigenvalue)

The complex eigenvalue is confusing me. $\lambda_{1,2} = \frac{1}{2}(5 \pm i\sqrt{3})$ for $$A=\begin{pmatrix}3&1\\-1&2\end{pmatrix}$$ so I get this for $\lambda_1 = \frac{1}{2}(5 - ...
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200 views

Find and draw the image of the triangle with vertices (2,1), (1, 2), (2,2)

a) Find the standard matrix of the linear transformation $T$, if $T:R^2 \rightarrow R^2$ reflects points through the line $y=x$ and then rotates points counterclockwise through π/4 radians. b) Find ...
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Eigenvalues of A when AA'A=A

In some presentation the speaker said that pushing AA'A-A to 0 makes encourages singular values of A to be either 0 or 1, can anyone tell me where this follows from?
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Spanning set of a sum of vector subspaces [on hold]

Let $A$ and $B$ be subspaces of a vector space $V$. Their sum $A+B$ is defined as: $$A+B= \{x+y \mid x\in A,y\in B\}$$ It's given that $A+B$ is also a subspace of $V$. Show that if $X$ is a ...
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63 views

Do we consider complex values for eigenvalues

I have an intro Linear Algebra assignment due tomorrow and I'm unsure of what the teacher expects. (I emailed him but he hasn't replied yet) Basically we never learned this stuff in class. It's an ...
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When calculating P, for diagonalization, does the order of eigenvalues matter?

I'm trying to find the value of this matrix, $A = \begin{pmatrix} 1 & 4 \\ 3 & 2\end{pmatrix}$ to the power of $10$. I've determined (and confirmed on Wolfram) that the eigenvalues are $5$, ...
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267 views

Intersection of lines in homogeneous coordinates

So yes, this is homework, but for the class that I'm TA'ing. The question is something like: Given two lines in 2D homogeneous representation ($\mathbb{R}^{2+1}$): $\bf a$ and $\bf b$, their ...
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New values of vector after change of base

We make a change of base with the matrix $$S=\left[\begin{matrix}p & q \\ 1 & 1\end{matrix}\right]$$ so the vector $x= (8, 3)$ of $\mathbb{R}^{2\times 1}$ becomes $x= (1, 2)$ and the vector ...
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Matrix representation of a transformation in a basis $B$

I need some clarification on this problem; my class notes and my current thought process are conflicting. I have a linear transformation $$T(a,b) = (a+2b, 3a-b)$$ and I'm asked to find $[T]_B$ ...
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103 views

Finding basis of a vector-subspace.

How can I get to know a basis of a vector-subspace of $\mathbb{R}^{2 \times 2}$ formed by matrices $X$ that commute with the matrix: $$A=\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right]$$ ...
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193 views

Finding an inner product for an orthonormal basis

I'm working through a question that I don't quite know how to get. It is: Find an inner product $\langle - , -\rangle$ on $\mathbb{R} ^3$, and a matrix A such that $\langle u, v \rangle = u^\top Av$ ...
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26 views

Functional equation with one given

If $f(0)=−10$ and $f(x)=(6x+4)^2−f(x+2)$ determine $f(3)$ I must be missing something. Thanks.
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250 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
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Find the Matrix A of the Linear Transformation

Can anyone walk me through the steps to complete this problem? I am unsure of where to start to solve the problem. I get that the resulting matrix $A$ should be a $2 \times2$ matrix, should I be ...
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244 views

Relationship between null space and invertibility of linear transformations

Is there a relationship between the null space $N(T)$ of a linear transformation $T$ and whether or not it is invertible? For example, if you know $N(T) \neq \{0\}$, can you be sure it's not an ...
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487 views

Is this an invertible linear transformation?

Suppose you have a linear transformation $T: M_{2\times 2}\to M_{2\times 2}$ given by $$ \begin{pmatrix} a & b \\ c & d\end{pmatrix}\mapsto \begin{pmatrix} a+b & a \\ c & ...
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254 views

Schur's complement of a matrix with no zero entries

Let $A$ be an $n\times n$ symmetric positive-definite matrix so that itself and its inverse $A^{-1}$ both have no entry equal to $0$ ($A_{i,j} \neq 0 $ and $(A^{-1})_{i,j} \neq 0$ for all $i,j \in ...
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120 views

Whether linear transformation maps $0$ to $0$

Suppose I have a linear transformation $T: V\to W$. If I perform this transformation on the $0$ vector of $V$, $0_V$, does that necessarily mean its image will be $0_W$? In other words, is it ...
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25 views

Diagonalizabilty of $A$

If $2$ is the only eigen value of $A\in Mat_{n\times n}(\mathbb C)$ then what can I say about the diagonalizabilty of $A$? I tried to check the equality of algebraic & geometric multiplicity of ...
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71 views

Let ${v_{1},v_{2},v_{3},v_{4}}$ be a basis of $\Bbb R^{4}$ and $v=a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3}+a_{4}v_{4}$

I came across the following problem: Let ${v_{1},v_{2},v_{3},v_{4}}$ be a basis of $\Bbb R^{4}$ and $v=a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3}+a_{4}v_{4}$ where $a_{i}\in \Bbb R,i=1,2,3,4.$ Then ...