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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Finding the transformation matrix of this linear map.

I've being doing several exercises and none was of this kind, which I can't figure out: Let $V$ and $W$ be vector spaces with basis $B=\{\vec{v_1},\vec{v_2},\vec{v_3}\}$ and ...
4
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1answer
73 views

Jordan's decomposition

I have a matrix $A\in R^{n,n}$. $A= \begin{bmatrix} 1&0&-2&0&0&\dots&0\\ 0&1&0&-6&0&\dots&0\\ 0&0&1&0&-12&\dots&0\\ ...
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2answers
84 views

A question about diagonalizable matrices

Let $A$ be a square matrix such that $A \ne0$, but $A^k=0$ for some integer $k \gt1$. show that $A$ is not diagonalizable. Could somebody give me some hints?Many thanks
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0answers
56 views

A question in linear algebra.

Let $A$ be a symmetric matrix with real coefficients. If $A^n=I$ for some integer $n\ge3$, prove that $A^2=I$ I have no idea in proving this question, hope somebody can help me.
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5answers
5k views

How to tell if two 3D vectors are in the same direction?

Given: $$AB=\left( \begin{array}{ccc}2\\1\\3\end{array} \right) \;\;\;\; \text{and}\;\;\;\; CD=\left( \begin{array}{ccc}4\\3\\6\end{array} \right).$$ Justify if $AB$ has the same direction as ...
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2answers
44 views

Prove the following

Prove that if $p$ and $q$ are polynomials over the field $F$, then the degree of their sum is less than or equal to whichever polynomial's degree is larger $$\deg(p+q)\leq \max \left\{\deg(p),\deg(q) ...
2
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1answer
34 views

Solve a system of three equations by rewriting in row-echelon form

I tried solving this system of equations and I got what seems to be an inconsistent system. I wanted to post my results here to see if I'm correct. Here is the original problem: $$ \left\{ ...
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1answer
39 views

Fractional and irrational matrix powers

What is a good reference to learn the basics about raising a matrix to a rational power or an irrational power? So I am interested in the existence and computation of things like $A^{\frac{1}{3}}$ or ...
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1answer
33 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
2
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1answer
895 views

Get the rotation matrix from two vectors

Given $v=(2,3,4)^t$ and $w=(5,2,0)^t$, I want to calculate the rotation matrix (in the normal coordinate system given by orthonormal vectors $i,j$ and $k$) that projects $v$ to $w$ and to find out ...
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0answers
99 views

Reference for Codimension in Infinite Dimensional Normed vector spaces

There are a couple identities I would like to use related to the codimension and its relationship to the annihilator; some of these seem to be true for all normed vector spaces, and others seem only ...
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1answer
98 views

Equations of planes and lines in 3-space

I'm reading Strang's book "Linear Algebra and it's applications" and he writes in the first chapter that an equation involving two variables in still a plane in 3-space. "The second plane is 4u - 6v ...
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1answer
147 views

How to write a linear map as a matrix with respect to a given canonical basis

I am asked to write a linear map as a matrix with respect to a given canonical basis. The basis is $b = \left \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right \} ...
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3answers
110 views

Solve this system by rewriting in row-echelon form $x+y+z=6$, $2x-y+z=3$, $3x-z=0$

This is my very first problem in Linear Algebra and I guess I really need to brush up on my Algebra skills..I'm at a loss as to how to solve this equation My reading said that there are basically 3 ...
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1answer
55 views

Is it true that positive definite matrices generates all the symmetric matrices?

Is it true that positive definite matrices generates all the symmetric matrices in $M_n(\mathbb{R})$? And is it true that the set of nonsingular symmetric matrices generates all the symmetric ...
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0answers
59 views

Verify: Orthogonal matrices are diagonalizable.

Verify: Orthogonal matrices are diagonalizable. I can't reach in any conclusion. All I can see is that an orthogonal matrix has $n$ linearly independent rows. However any nonsingular matrix has the ...
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1answer
22 views

Recovering $X,Y,Z$ from $x,y,z$ in CIE color model.

The background of where I'm getting this from is less important, but you can read it if you like. The question is, given three variables $X$, $Y$, and $Z$, all ranging from $0$ to $1$, we can ...
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1answer
104 views

Equation of a line through a point that intersects two crossing lines.

Find the equation of a line through a point, $P(7,1,1)$, that intersects two crossing lines $a$ and $b$. Where $$ a\;\begin{cases}2x+z&=0\\2x-y-1&=0\end{cases} \quad\text{ and }\quad ...
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0answers
89 views

Moore-Penrose pseudoinverse of a 3×3 matrix

Is there a "simple" formula for computing the Moore-Penrose pseudoinverse of a $3\times 3$ matrix? I mean something like the formula for the inverse (for non-singular matrices), which involves the ...
2
votes
1answer
260 views

Derivative of $(Ax - b)^T(Ax-b)$

I am trying to take the derivative of $(Ax - b)^T(Ax-b)$ and setting it to zero without expanding the multiplication, by only using matrix calculus. I knew the partial derivative of $x^Tx$ according ...
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1answer
91 views

Dual and Second Dual Basis

Let $B={e_1, e_2, e_3 }$ the canonical basis of $\mathbb{R}^3$. Build the dual and second dual basis of $\mathbb{R}^3$. This is a question about finding the base to a vector space which makes a ...
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1answer
89 views

Linear operator $T^k$ effect on $kerT^k$ and $ImT^k$

Let $T:V\rightarrow V$, an linear operator. In general, what can you say about $kerT^k$ and $ImT^k$? For example, I've understood that $kerT^{k-1} \subseteq kerT^k$. I'd like to know what else ...
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1answer
673 views

Given an implicit 3D plane, how do I find the orthogonal projection matrix - which projects any point - onto this plane?

The plane is given by the equation $Ax+By+Cz+d = 0$. Can you tell me how can I figure out the 4x4 matrix which orthogonally projects any point given by homogeneous coordinates onto this plane? I am ...
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0answers
37 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
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1answer
66 views

How to prove this by mean value theorem? $f(y)=f(x)+\nabla f(x)^T(y-x)+\frac{1}{2}(y-x)^T\nabla^2f(x+a(y-x))(y-x)$

How to prove this by mean value theorem? $f(y)=f(x)+\nabla f(x)^T(y-x)+\frac{1}{2}(y-x)^T\nabla^2f(x+a(y-x))(y-x)$ where $a\in[0,1]$. The mean-value theorem is $\frac{f(y)-f(x)}{y-x}=\nabla ...
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2answers
38 views

Does conjugation preserve spectrum of matrices?

Actually, I saw normalizer of diagonal matrices are permutation matrices. I read the answer but I don't know how to prove that conjugation preserves the spectrum. Actually I do some proof on 2x2 ...
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2answers
77 views

Universal property of tensor products / Categorial expression of bilinearity

Let $V$ and $W$ be linear spaces. According to Wikipedia, the universal definition of the tensor product $V \otimes W$ satisfies the following property: There is a bilinear map (i.e., linear in each ...
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0answers
34 views

Calculating with transformation matrix

Given is the transformation of coordinates $ T_{AB} = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} $. 1.) What are the new coordinates for the vectors (1,0) and (0,1)? It should be: $ ...
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1answer
49 views

Equivalence of Lattices

Let $\Gamma=\{mw_1+nw_2:m,n\in\mathbb{Z}\}$ and $\Gamma'=\{mw_1'+nw_2':m,n\in\mathbb{Z}\}$. Show that $\Gamma=\Gamma'$ if and only if there exists a matrix $A\in SL(2,\mathbb{Z})$ such that $\left( ...
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1answer
27 views

What is the apex for the parabola $y^2+py=px-2p$?

Formula: $y^2+py=px-2p$ For which value (s) of $p$ is the apex of the parabola on the line $y = x$ is the parabola at the right side of the $y$-axis? $y^2+py=px-2p$ can be written as ...
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1answer
70 views

Artin's proof of linearity of determinant in rows of matrix

Definition of linearity: Let $A_i$ denote the $i$th row of matrix A. Let $A, B, D$ be matrices, all of whose entries are equal except for those in row k. Suppose furthermore that $D_k = cA_k + c'B_k$ ...
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1answer
539 views

Solving differential equations in linear algebra

I'm having a hard time early on in this linear algebra course, I'm a first year student in University. I'm reading my textbook right now and it gives the following differential equation as an example ...
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1answer
297 views

Calculate equal distance between lines and points

How do I do something like this?: Consider the lines of k: x = 4 and l: y = 4x + 2, and the point A (0, 6). What is the equation of the parabola 'p' with focus 'A' and directive k? And calculate ...
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2answers
765 views

Given the following vector $X$, find a non-zero square matrix $A$ such that $AX=0$:

Given the following vector X, find a non-zero square matrix $A$ such that $AX=0$: So this problem stumped me and I've resorted to stack exchange. I need to find $A$, when I have a vertical vector ...
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3answers
68 views

Prove this is a subspace

Let $ W_1, W_2$ be subspace of a Vector Space $V$. Denote $W_1+W_2$ to be the following set $$W_1+W_2=\left\{u+v, u\in W_1, v\in W_2\right\}$$ Prove that this is a subspace. I can prove that the ...
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1answer
35 views

Parallel Lines, One point on each.

If I have two parallel lines, and I know only 1 point on each, is it possible to calculate their slope or any other information about them? Thanks
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1answer
45 views

Planes in linear algebra

I have a question, say I have two linearly independent vectors, then would there be only one plane in R3 containing these two vectors?
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2answers
123 views

Solving homogeneous systems using Gaussian elimination

I have a system of equations: $$2x_1 + 6x_2 - 4x_3 = 0$$ $$3x_1 + x_2 + 7x_3 = 0$$ $$4x_1 - x_2 + 2x_3 = 0$$ I have tried to solve it, but I'm stuck at this part: ...
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2answers
36 views

A question about eigenvalues

Let $v=\begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix}$ be a nonzero column vector in $\Bbb R^4$ and let $A=vv^T$. Find the eigenvalues of $A$ There must be a easier way rather than calculate it ...
2
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2answers
113 views

Understanding Gauss-Jordan elimination

I have a following system: $$x_1 + x_2 - x_3 = 5$$ $$2x_1 + 2x_2 - 4x_3 = 6$$ $$x_1 + x_2 - 2x_3 = 3$$ I dont understand how to solve this system using Gauss-Jordan elimination. I was told it I had ...
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19answers
9k views

What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
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3answers
212 views

Matrices - Understanding row echelon form and reduced echelon form

I have the following two matrices: 1) $$\begin{bmatrix}1&0&0\\ 0&1&1\\ 0&0&0\\0&0&0 \end{bmatrix}$$ I believe this matrix is in the form of reduced row echelon form ...
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2answers
257 views

Intersection of two spans

Let $\mathrm{span}\{ {v_1}...{v_j}\} \cap \mathrm{span}\{ {v_{j + 1}}...{v_n}\} \ne \{ 0\} $. So, there's a vector, not $0_v$ in the intersection. Why does it imply that there exist $a_i, b_i$, ...
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1answer
91 views

Nilpotent matrix in $\mathbb R$

We can prove that $A \in \mathbb C^{n,n}$ is nilpotent ($\exists m\ A^m=0$) if $p_A(t)=t^n$, where $p_A$ is the characteristic polynomial of matrix $A$. What if $A \in R^{n,n}$? Proof that ...
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1answer
166 views

How do I row reduce a matrix mod 26 when it is singular mod 26?

Cryptography assignment question: matrix $A$ is \begin{equation} A = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 3 & 1 \\ 0 & 2 & 5 ...
2
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1answer
111 views

Change of basis- vector space of polynomials

Given is the vector space of Polynomials of degree $\le3$ and the basis $$\mathcal{B}_2 = \left\lbrace1,x-1,x^2-3x+2,x^3-6x^2+11x-6\right\rbrace$$ Furthermore the linear mapping $$L: p(x) \rightarrow ...
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1answer
42 views

Rotation on extension-field

I have corrected the question in the following. For $x_1$ and $x_2$ real vectors which span $V=\mathbb{R}x_1\oplus \mathbb{R} x_2$. we have a rotation $R$ on $V$ given by \begin{eqnarray*} ...
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2answers
396 views

Raising $e$ to the power of a matrix

Does there exist a definition for matrix exponentiation? If we have, say, an integer, one can define $A^B$ as follows: $$\prod_{n = 1}^B A$$ We can define exponentials of fractions as a power of a ...
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1answer
63 views

Finding an integer orthogonal basis

Say I have some non-orthogonal basis of some vector space that only have integer elements. Is it possible to find an orthogonal basis consisting of basis vectors with integer elements?
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1answer
39 views

finding this linear transformation

i am following this guide: http://www.calpoly.edu/~brichert/teaching/oldclass/f2002217/handouts/goof.pdf my question is to find the linaer transformation that adheres to $T(1,1,1) = (1,1,1)$ ...