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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1answer
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Find the $n^{th}$ power of a $2$x$2$ matrix.

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. Using Lagrange's interpolation compute $A^n$ for $n\in\mathbb{N} $ So far I've worked out the minimum polynomial of $A$ to be $(x-2)(x+1)$ but ...
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4answers
91 views

Why is the volume of a parallelepiped equal to the square root of $\sqrt{det(AA^T)}$

Why is the $\sqrt{det(AA^T)}$ equal to the volume of a parallelepiped? Is is somehow related to the fact that $det(A) = det(A^T)$? EDIT: To clarify, the parallelepiped is spanned by the columns of ...
2
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2answers
33 views

Solving Simultaneous Equations - Hill Cipher

I have searched but an unable to find any examples like what I am faced with. Plaintext = SOLVED CipherText = GEZXDS 2x2 encryption matrix $$ \left(\begin{matrix} 11 & 21 \\ 4 & 3 ...
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1answer
24 views

Prove the span of a set of vectors is a subspace. [on hold]

If $v_1,v_2,v_3,...,v_p \in V$, where $V$ is the vector space, then span ${v_1,v_2,v_3,...,v_p}$ is a subspace of $V$. Is there a proof available that could help me better understand this?
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13 views

Invent a linear mapping given the following conditions.

Invent a linear mapping L such that: $L(1,2)=(3,5)$ and $L(-2,1)=(2,-3)$ I'm just unsure on how to start this problem, if anyone can give me any tips that be great thanks!
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0answers
14 views

Finding a basis of a subspace [on hold]

Let V = $P(Z_{5}).$ (a) Let S $\subset$ V be given by S = $(x + 1, x^2 + x, x^2 + 2, x^2 + x + 1)$ Find a subset S' of S which is a basis for sp S.
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1answer
22 views

How does the hermitian conjugate of an unitary operator act?

I know that $$ A| v \rangle = \sum _n e^{i\alpha n} | n \rangle $$ where $A$ is an unitary operator, and $ \left \{ |n\rangle \right \} $ is an orthonormal complete basis. In that case, is it true ...
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1answer
44 views

What is the dimension of the space $V$ of all matrices $S$ [on hold]

Matrix $A$ represents the orthogonal projection onto a plane $R \in \mathbb{R^3}$. What is the dimension of the space $V$ of all matrices $S$ such that: $$AS = S \left( \begin{array}{ccc} 1 & 0 ...
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0answers
19 views

Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by u, v and w. [on hold]

Consider the vectors $u= (-2, -2, 2)$, $v=(-1, 2, -3)$ and $w=(-6, 0 -2)$. Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by $u$, $v$ and ...
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2answers
19 views

beginner with linear approximations

So I am just learning about linear approximations in my class. But I just don't understand how to figure out the answer. Can someone please explain it in more detail to me thanks.
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1answer
23 views

Verify that $p \circ p=p$ but that $p$ is not self-adjoint.

Consider the following subspaces of $\mathbb{R}^2$: (a) $\{(x,0)\}$ (b) $M=\{(x,y):x=2y\}$. Find an expression and a matrix representation for the oblique projection $p$ of a vector $v \in ...
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0answers
16 views

If $\mathcal{L}(V)$ is the set of all functions $T:V\rightarrow \mathbb{R}$, then $\mathcal{L}(V)$ is a vector space

Suppose $V$ is a vector space, and $\mathcal{L}(V)$ is a set of all functions $T:V\rightarrow \mathbb{R}$ such that $T(c_1f_1+c_2f_2)=c_1T(f_1)+c_2T(f_2)$. Show that $\mathcal{L}(V)$ is a ...
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1answer
33 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
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1answer
14 views

Let $M=\{(x,y,z):z=3x-y\}$. Find the orthogonal projection of a vector $v \in \mathbb{R}^3$ on $M$ and a matrix representation of $p_M$.

Let $M=\{(x,y,z):z=3x-y\}$. Find the orthogonal projection of a vector $v \in \mathbb{R}^3$ on $M$ and a matrix representation of $p_M$. I know that the orthogonal projection of two vectors is ...
2
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1answer
28 views

What's the difference between superscripts and subscripts on fields?

$P_2$ vs $\mathbb{R}^3$ $P_2$ represents the set of polynomials equal to or less than degree 2. $\mathbb{R}^3$ represents (I think) a vector with 3 elements. Is there any difference if the number ...
5
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1answer
128 views

Symmetrical of a triangle's vertexes

I have the following problem : Show that the symmetrical (ie reflection) of a triangle's vertexes by the opposite side are aligned iff the distance between the orthocenter and the circumcenter is ...
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2answers
15 views

Prove that $p_M+p_{M^\perp}=I$, where $I$ is the identity on $H$.

$M$ is a subspace of $H$, a vector spaces of finite dimension with an inner product $\langle \rangle$; $p_M$ is the orthogonal projection on $M$ and $M^\perp$ is the orthogonal complement of $M$. ...
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1answer
22 views

Find a tight frame in $\mathbb{R}^2$ with 10 elements of lengths $\{ \mathscr{L}_1,\dots,\mathscr{L}_{10} \}$, with

Find a tight frame in $\mathbb{R}^2$ with 10 elements of lengths $\{ \mathscr{L}_1,\dots,\mathscr{L}_{10} \}$, with $\mathscr{L}_1=1$, $\mathscr{L}_2=2$, $\dots$, $\mathscr{L}_{10}=10$. Is it ...
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2answers
24 views

what is the null space of a onto linear transformation?

i don't know if im conceptually understanding null space. For a one to one the only vector in the null space has to be zero.is that the case for an onto transformation?
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0answers
13 views

Coordinates of derivatives in the standard polynomial basis

Consider the polynomial $f(x)=x^5-5x^4$. Find coordinates of $f',\ f''$ and $f'''$ in the basis $\{1,\ x,\ x^2,\ x^3,\ x^4,\ x^5\}$ What I have so far is to take the derivative say ...
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1answer
108 views

Solving non-square linear systems with the exterior product and Cramer's rule

I'm reading the book Linear algebra via exterior products by Sergei Winitzki (which is the worst book, ever) and he shows that you can solve linear systems with a general solution with Cramer's rule ...
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1answer
106 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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0answers
27 views

Linear Algebra Practice T/F [on hold]

Here's a list of practice problems from my Linear Algebra teacher. We're suppose to determine if they are true or false. Explanations are greatly appreciated. If A is nxn and nonsingular. then A^2 ...
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1answer
25 views

Rank of tensors in terms of ranks of associated linear maps

Let $V$ be a vector space over a field $k$, let $w \in \otimes^l V$ be a tensors. We call $w$ a simple tensor if it can be written as $$ w=w_1 \otimes w_2 \otimes \ldots \otimes w_l, $$ where $w_i \in ...
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2answers
22 views

What is the difference between a linear independent set and a generating set?

im having difficulty because onto have columns of generating set and one to one has columns of linear independence but the way we prove whether a standard matrix has linear independent columns are ...
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0answers
104 views

Homology out of Smith normal form: simultaneous or independent diagonalization?

Let $R$ be a PID and $R^m\overset{A}{\longrightarrow} R^n\overset{B}{\longrightarrow} R^o$ matrices with $BA=0$ and Smith normal forms ...
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1answer
26 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
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68 views

Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
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1answer
65 views

If $\Phi: \mathbf{Vec} \rightarrow \mathbf{Vec}$ with $\Phi(V) = V^{\ast\ast}$ and $f: V \rightarrow W$, what is $\Phi(f)$?

Let $\Phi$ be an endofunctor of the category of vector spaces over a field which sends a vector space to its double dual. Let $V$ and $W$ be 2 vector spaces and let $f: V \rightarrow W$ be a morphism ...
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1answer
14 views

Linear Independence Solving for $h$

For which values of h will the vectors $v_1 = (h , -1/2, -1/2)$, $v_2 = (-1/2, h, -1/2)$, $v_3 = (-1/2, -1/2, h)$ form a linearly dependent set? There are apparently two answers for $h$ here. I ...
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0answers
64 views

Making a complex inner product symmetric

Let $(V, (\cdot, \cdot))$ be a complex inner product space, say a space of complex-valued functions, with $(\cdot, \cdot)$ linear in the second position and sesquilinear in the first. Assume that $V$ ...
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0answers
22 views

Representing Vectors as Matrices

Is there any good reason why the matrix representation of a 3-D vector should be of the form $$( \ \ \ \ z \ \ \ \ \ \ x - iy) \\ (x + iy, \ \ - z \ \ )$$ A few reasons would be even better! e.g. ...
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1answer
32 views

determinant of a linear map

When we define the determinant of a linear transform $T:V\rightarrow V$ we consider it to be the determinant of the matrix we obtain as a matrix representation of $T $ corresponding to a given basis ...
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1answer
243 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
1
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1answer
36 views

What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$?

What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$? I want to say that it is at least $2^{\aleph_0}$, but I have no idea how to sharply pin it down otherwise.
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1answer
22 views

Norm of the sum of inverse matrices

Let $A,B$ be two invertible matrices. Is there a way to compute $\|A^{-1} -B^{-1}\|$ in terms of $\|A-B\|$?
3
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1answer
32 views

Notation in formula for tensor product of Hadamard matrix

I'm having trouble understanding the notation used in a linear algebra exercise (it's exercise 2.33 of Nielsen and Chuang's "Quantum Computation and Quantum Information", page 74). The exercise gives ...
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1answer
15 views

Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
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0answers
28 views

the set of points equidistant from $ u $ and $v$ form a line.

Let $u$ and $v$ be two vectors in $ \mathbb{R}^2 $ with the standard norm. Show that the set of points equidistant from $ u $ and $v$ form a line. I show that if $x$ is equidistant from $u$ and $v$, ...
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1answer
25 views

Element matrix multiplication representation

Matrix element by element multiplication defined : $C=A*B$ $c_{ij}=a_{ij}b_{ij}$ Is this multiplication can be represented with stardant matrix multiplication or Kronecker product ?
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Is there a name for the followin 'norm' on vector spaces?

Let $V$ be a real vector space of dimension $n$. Define a "norm" by $$\|v\| = \frac{\sum v_i}{n} - \min v_i.$$ So the "norm" is the difference between the average value and the minimum value. It is ...
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2answers
248 views

Linear transformation is invertible if and only if $x$ does not divide its minimal polynomial

Let $T : V \to V$ be a linear transformation of a finite dimensional vector space over a field $\mathbb{F}$ to itself. Prove that $T$ is invertible if and only if $x$ does not divide the minimal ...
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0answers
15 views

Let f : I → J be a function between sets I and J…

Let f : I → J be a function between sets I and J , and let F be a field. Show that there is a unique linear map φf : FI → FJ such that φf (ei) = ef(i). Deduce that if f is the identity map on I then ...
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1answer
39 views
+50

Find the eigenvalues of the matrix and give the bases for each of the corresponding eigenspaces

I'm having issues with this problem. I'm getting really weird eigenvalues and then I have no clue how to solve the matrix to get the bases. I have to follow this method. Below are the pictures of my ...
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1answer
28 views

Is $vv^{T} - v^{T}vI$ non-singular? [on hold]

Is $vv^{T} - v^{T}vI$ non-singular ? Why? $v$ is vector
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1answer
27 views

What are the possible dimensions of all $n \times n$ matrices such that $A\vec{x} = \vec{0}$?

If $\vec{x}$ is any vector in $\mathbb{R^n}$, what are the possible dimensions of the space $V$ of all $n\times n$ matrices $A$ such that $A\vec{x}=\vec{0}$? I'm confused about a couple of ...
0
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0answers
18 views

By Quadratic Equation [on hold]

It takes Meihua two hours more to complete a 50km journey than it takes Allin to complete a 40km journey. If the average speed of Meihua for the journey is 5km/h less than Allin, calculate the average ...
3
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0answers
42 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
5
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1answer
53 views

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable (Tensors, or k-linear forms)

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable. $f\in A_k(V)$ is decomposable if there exists a $a_1,...,a_k\in V^\wedge$ such that $f=a_1\wedge...\wedge a_k$ In this case "let ...