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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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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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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665 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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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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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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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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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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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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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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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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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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529 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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291 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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761 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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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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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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121 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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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 ...
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112 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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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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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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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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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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165 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 ...
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1answer
109 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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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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395 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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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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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 $T(1,1,1) = (1,1,1)$ ...
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1answer
68 views

Least-squares matrix form

I was reading through linear regression but I cannot get my head around with the notation. Given a set of points $(x_1, y_1), \ldots, (x_n,y_n) \in \mathbf{R}$ the least-squares approximation is can ...
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118 views

Prove vector spaces dimensions equality

Let $V$ a finite vector space over $F$, and $W$ a vector space over $F$ with the dimension of $1$. $S:V\rightarrow V, T:V\rightarrow W$ two linear transformations. It is given that $\ker S$ is not ...
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4answers
64 views

A quadratic function is a function of the form $y=ax^2+bx+c$, where $a, b$ and $c$ are constants.

Given any 3 points in the plane, there is exactly one quadratic function whose graph contains these points. Find the quadratic function whose graph contains the points $(−5,112), (0,2),$ and ...
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404 views

If augmented matrix has zero row, does it mean infinite solutions?

Given an augmented matrix in REDUCED ROW ECHELON FORM representing a general system of linear equations (does not have to homogeneous), if the augmented matrix has a zero row, does it imply that the ...
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Prove that $n$ is even and $|A| \in \{-1,1 \}$

Let $A \in M_{n} (\mathbb R)$, such that $A^2=-I_{n}$. Prove that $n$ is even and $|A| \in \{-1,1 \}$. I started by compute the determinant of both sides: $A^2=-I_{n}\Leftrightarrow$ ...
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60 views

A question in linear transformation

Let $V$ be a finite dimensional vector space and $T:V \to V$ be a linear transformation. Prove there exists a linear transformation $S:V \to V$ such that $TST=T$ I feel this should not be a hard ...
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133 views

A vector is a linear combination of two other vectors. Find all scalars c.

Q: The vector $[13, -15]$ is a linear combination of the vectors $[1, 5]$ and $[3, c].$ Find all scalars c. My Approach: $$[13, -15] = x[1, 5] + y[3, c]$$ $$13 = x + 3y \to (1)$$ $$-15 = 5x + cy ...
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1answer
110 views

How unique (on non-unique) are U and V in Singular Value Decomposition (SVD)?

According to Wikipedia, "A common convention is to list the singular values in descending order. In this case, the diagonal matrix $\Sigma$ is uniquely determined by $M$ (though the matrices $U$ and ...
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0answers
29 views

Intersection of affine subspace of $\mathbb{R}^n$ with $[0, 1]^n$

Suppose I have an affine subspace $V \subseteq \mathbb{R}^n$, say given by a rank-$r$ system of $m$ equations in $n$ variables. I'm interested in two questions: Is there a straightforward way to ...
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1answer
116 views

Find an upper triangular matrix

Let $T:V\rightarrow V$ such that: $T(a+bx+cx^2) = (a+b) + (b-c)x + (a+b+c)x^2$ $B=\{1,x+1, x^2+1\}$. Find a basic $C$ such that ${[T]_{B \to C}}^{}$ is an upper triangular matrix. Use $T$ for ...
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Do linear transformations on curves defined by polynomials preserve tangents?

Suppose we have a curve $C$ defined by $F(X,Y) = 0$ where $F$ is a polynomial. Suppose $l$ is a curve that is tangent to $C$ at some point $P$ and that $M$ is an invertible linear transformation ...
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54 views

Symmetric $\mathbb F_2$-bilinear form

$K$ is a field with $2^d$ elements. $K^*$ is a cyclic group generated by $\alpha$ and $T:K\rightarrow \mathbb F_2$ is a non-zero $\mathbb F_2$ linear map. Can you help me proving that the symmetric ...
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99 views

If the vector space of all real valued continuous functions on the metric space (X,d) is finite dimensional then X is finite set

If $(X,d)$ is a metric space such that $C(X,R)$ is a finite dimensional real vector space, would any one help me to show that $X$ is finite set? $C(X,R)$ denotes the set of all real valued continuous ...
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115 views

Given a matrix of basis transformation what is the algorithm to find $ker(T)$ and $im(T)$?

I'm given the following transformation matrix of the linear map $T:\mathbb R^4\to\mathbb R^3$: Find $\mathrm{ker}(T)$ and $\mathrm{im}(T)$ So I should probably get this matrix to $\mathrm{rref}$: ...
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28 views

If $λ_i > 0, \forall i$, $A$ is positive definite

Given that $A \in R^{n,n}$, $λ_i $ the eigenvalues and $x_i$ the eigenvectors ($x_i^Tx_j=δ_{ij}$). I have to show that if $λ_i > 0, \forall i$, $A$ is positive definite. My idea is the following: ...
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Solving an equation with the form $Ax=b$

$$\begin{array}{l} \left( \begin{array}{l} \begin{array}{*{20}{c}} 1 & 2 & 3 & \cdots & n \\ \end{array} \\ \begin{array}{*{20}{c}} 2 & 3 & 4 & \cdots & ...
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1answer
296 views

Commuting matrices and simultaneous diagonalizability

It is a known fact from linear algebra that if a set of matrices is pairwise commutable then they are simultaneously diagonalizable. A problem in the book I am currently studying asks to prove this ...
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Finding the basis of $U=\{ A\in M_{2\times 3}(\mathbb R) |a_{11}+a_{21}+a_{22}+a_{23}=0 \}$

Find the basis of $U=\{ A\in M_{2\times 3}(\mathbb R) |a_{11}+a_{21}+a_{22}+a_{23}=0 \}$ So we set $a_{11}=-a_{21}-a_{22}-a_{23}$ Then why is this a basis ? I get that its dimension should be 5 ...
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55 views

Prove that vectors x,y are linearly dependent exactly when …

Prove that vectors $\vec{x},\vec{y}$ (belonging to $\mathbb{R}^3$) are linearly dependent only if the following is true $$ \begin{vmatrix} x_1&y_1 \\ x_2&y_2 \end{vmatrix} ...
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1answer
49 views

Extreme of function - geometry

Hi I am helping with one homework assignment. Two corridors of width 320 cm and 135 cm are intersecting each other with angle of 90 degrees. Find the maximum length of thin non-flexible (unbending) ...