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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What is the base and dim for the kernel of this linear transformation

Ok, so i have a linear transformation that is from second degree polynomial to a $2\times 2$ matrix $$T : \mathbf{P_{2}[X]} \to \mathbb{R^{2x2}}$$ which defined as: $$T(P(X)) = \begin{pmatrix} P(1) ...
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17 views

Decomposable polynomails of 2 degree in finite space $\mathbb{F}[X]$

How can I show that there is a decomposable polynomial of second degree in a finite space $\mathbb{F}[X]$? I tried contrapositive proof but I got stuck. That made me think that maybe I should go for ...
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0answers
924 views

How to convert from cartesian to vector form (of a straight line)

line in question: $$-x - 1 = \frac 12y - \frac12 = \frac 12z + 1$$ I wasn't really sure how to go about this one since it's not in the exact general form that I was taught, but I went along and ...
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1answer
50 views

Jordan Canonical Forms: Different Approaches

Let $\dim(V)=n$ over the field $\mathbb{C}$. The Jordon canonical form of a linear transformation $T\colon V\rightarrow V$ can be obtained in the following way. 1) Let ...
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2answers
155 views

Under what conditions is the product of two invertible diagonalizable matrices diagonalizable?

The answer in this question gives an example for the statement product of two invertible diagonalizable matrices is not diagonalizable. My question is: Are there some conditions, perhaps ...
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3answers
148 views

Determinant of matrix times a constant.

Prove that $\det(kA) = k^n \det(A)$ for and ($n \times n$) matrix. I have tried looking at this a couple of ways but can't figure out where to start. It's confusing to me since the equation for a ...
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1answer
63 views

Optimal solutions of x and y for $\max_{x,y}~\min (f(x,y),~g(x,y))$

Can someone help me to find analytical solutions for optimal values of $x$ and $y$ which satisfy the following optimization problem? \begin{align} \max_{x,y}~\min & ...
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1answer
21 views

Determinant question about exchanging rows & columns.

Let A be a $4 × 4$ matrix, and choose $i, j, k, l \in [4]$ with $i \ne j$ and $k \ne l$. Let $B$ be the matrix obtained from $A$ by exchanging the $i^{th}$ and $j^{th}$ rows, and by exchanging the ...
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0answers
43 views

Solve the following system of linear ODE

$$x'_{1}=x_{1}-x_{3}$$ $$x'_{2}=4x_{1}-3x_{2}-x_{3}$$ $$x'_{3}=x_{1}+x_{3}$$ What i tried I first convert it to a system of 3x3 matrix $$ A=\begin{bmatrix} 1 & 0 & -1 \\ 4 & -3 & ...
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32 views

Clarification on Homomorphism and Automorphism of Vector Spaces

I've been struggling to connect some of these concepts for a while now and seem to be confusing myself more than helping myself by continuing to think about them. Could someone confirm or deny my ...
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1answer
135 views

Orthonormal and/or Orthogonal Basis of a Pair of Vectors

I was hoping someone could verify if this is the correct way to answer this problem: Let $\mathbb{R^{2}}$ have the standard dot product. Classify the following pair of vectors as (i) basis, (ii) ...
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39 views

Prove that T1+T2 and cT1 are linear transformations

Sorry to ask two questions in a day, but I was struggling with this problem. I'm probably overthinking it. If $T_1$ and $T_2$ are linear transformations from V into W, verify that $T_1+T_2$ and ...
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47 views

What are the objects and morphisms of the category $\operatorname{Vect}$?

What are the objects and morphisms of the category $\operatorname{Vect}$? I am trying to learn category theory, and it seems we have infinite objects in $\operatorname{Vect}$ being all of the finite ...
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2answers
38 views

Matrix computing of $a(i,j)a(j,i)$

I have a square, semi-positive matrix $A$ and I want to compute the sum of the products $a(i,j)a(j,i)$ for every $i$ and $j$. Is there any easy way to perform this computation that does not involve ...
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1answer
94 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
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68 views

Does in this case exist necessarely an eigenvalue equal to $0$?

I pasted more than I refer, hoping to be more clear. Look at the claim of the theorem: it states we can change coordinates untill we reach a "good" form for the equation of $r$, which defines the ...
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1answer
28 views

Projections onto Matrices

I just need some guidance whether I have the concept down. Consider a $2\times 2$ matrix together with the inner product $\langle A, B\rangle= \operatorname{Trace}(A^TB)$, and let ...
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1answer
58 views

Struggling with Differential Operators?

I'm taking a basic linear algebra/Differential Equations class hybrid (weird right?), and we're currently learning about differential operators. Am I correct in saying that a differential operator is ...
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1answer
24 views

Vector inequality for a scalar difference of two vectors in $\mathbb{R}^n$.

A student posed an interesting problem to me the other day and embarrassingly I could not prove or disprove it even though it appears relatively simple. The question was: Given vectors ...
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4answers
229 views

How to prove that ${1,x,x^2}$ is a basis of a real polynomial functions space.

Let $V$ be the real vector space of all polynomial functions from $\mathbb{R}$ to $\mathbb{R}$ at most second degree. That is, the space of all functions with form $f(x)=c_0+c_1x+c_2x^2$ with ...
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1answer
52 views

Linear Algebra: Isomorphism Proof

Here are the initials: $\nu$ is a vector space $\beta=\begin{Bmatrix}b_{1},\cdots,b_{n}\end{Bmatrix}$ is a basis for $\nu$ Argument: coordinate mapping $\Phi_{\beta }:\nu \rightarrow ...
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209 views

Prove the distributibe property of the dot product using its geometric definition?

The geometric definition of the dot product: $$\mathbf{a} , \mathbf{b} \in \mathbb{R} ^n$$ $$\mathbf{a} \cdot \mathbf{b} = a \cdot b \cos \theta$$ The distributive property of the dot product: ...
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1answer
40 views

Showing that a basis is an orthonormal basis with respect to given inner product

I'm having difficulty showing that a basis is an orthonormal base for a vector space. The exercise is as follows: Let $V$ be a real or complex vector space (possibly infinite-dimensional), and let ...
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32 views

projection and self-adjoint matrix

"Let $V$ be an inner product space that is an inner direct sum $U_1 ⊕ U_2$ of its subspaces $U_1, U_2$. Let T : V → V be the projection to $U_1$ along $U_2$. Prove that $T$ is self-adjoint if and only ...
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2answers
292 views

Proof of identity: cross product of three vectors

A book I'm reading contains the following (paraphrased) \begin{equation} (a \times b) \times c = (a \cdot c)b - (b \cdot c)a \end{equation} This is supposed to follow from: \begin{equation} (a \times ...
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2answers
62 views

Quadratic Forms and Associated Matrices

This might be a dumb question but when we write the matrix associated with a quadratic form, why does the matrix need to be symmetric in general? I'm asking because I'm thinking there isn't a unique ...
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2answers
27 views

adjoint map of differentiation space

"Let $V$ be the vector space of real-valued infinitely differentiable functions $f$ on R that are periodic with period 1, that is, $f(x) = f(x + 1)$ for all $x ∈$ R. Consider the inner product ...
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1answer
49 views

how to prove that a matrix is definitely invertible

T $\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)$ = $\bigl( \begin{smallmatrix} a & -b \\ b & a \end{smallmatrix} \bigr)$ prove that any given matrix on image of ...
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1answer
112 views

Prove that rref[A|AB] = [I$_{n}$|B] for an invertible matrix A, and some arbitrary matrix B

I am trying to prove that rref[A|AB] = [I$_{n}$|B], given an invertible matrix A and another matrix B. Note, B does not have to be invertible, but both A and B are n x n matrices. I understand that ...
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37 views

Qualitative properties of eigenvalues that can be inferred from matrix structure?

I am doing a linear stability analysis of a 6-dimensional system, what I want to know is if the system is stable at numerically solved steady states by looking at the eigenvalues of the jacobian ...
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1answer
72 views

Linear Algebra. Norm.

If $\|v\| = 2$ and $\|w\| = 3$ , what are the largest and smallest values possible for $\|v-w\|$ ? Give a geometric explanation of your results.
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44 views

Let $\operatorname{Rank}(A)=k$. Why must some $k$-by-$k$ submatrix of $A$ have nonzero determinant?

Let $\operatorname{Rank}(A)=k$. Why must some $k$-by-$k$ submatrix of $A$ have nonzero determinant? And why does every $(k+1)$-by-$(k+1)$ submatrix of $A$ have zero determinant?
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4answers
60 views

If $ax + by = a(b-1) + b(-1)$, then does $x = b-1$ and $y = -1$

In this case, $x$ and $y$ are variables and $a$ and $b$ are arbitrary constants. It seems like just looking at the equation that this would be true, but is there a case when it does not work? If I try ...
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0answers
38 views

Problem with the primary decomposition theorem

I need to use the primary decomposition theorem in the linear transformation $T:\mathbb R^3 \to \mathbb R^3$ defined by the matrix $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & ...
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1answer
117 views

A question on “Linear Algebra” by Kenneth Hoffman

I'm reading "Linear Algebra" by Kenneth Hoffman and Ray Kunze. I'm now lost at $\S$6.4 Theorem 6: the proof looks OK, but when I pick an example, somehow it does not tally. Please find below the ...
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2answers
32 views

Linear subspace closed under all special orthogonal matrices

Let $n\in \mathbb N$ and $E$ be an $n$ dimensional vector space over $\mathbb R$. Let $F$ be a linear subspace of $E$ such that $\forall f\in SO(E), f(F)\subset F$ Prove that $F=\{0\}$ or ...
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1answer
55 views

Characteristic polynomial of differential of some function

Let $E$ be the four-dimensional real vector space $M_{2\times 2}$ of real $2\times$2 matrices. Show that by setting $$f(X)=X^2$$ for 2$\times$2 matrix $X$, we define a continously differentiable ...
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3answers
322 views

Does $\{y\in \mathbb{R}^n:\operatorname{rank}((x,y,Ay))=2\}$ have zero Lebesgue measure?

This is probably a simple question, but I need some help. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ matrix $A$. I'm interested in the set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ ...
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2answers
60 views

$W$ is a subset of $\mathbb{R}^3$ defined as $W = \{(x, y, z)|x + y + z ≥ 1\}$. Is $W$ closed under vector addition and scalar multiplication?

$W$ is a subset of $\mathbb{R}^3$ defined as $W = \{(x, y, z)|x + y + z ≥ 1\}$. (a) Is $W$ closed under vector addition? If your answer is no, then find two vectors $u, v \in W$ such that $u + v ...
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1answer
109 views

let $\dim(ker (A - \lambda I)) = 1$. why is $adj(A - \lambda I) \ne 0$

Let $\lambda$ is eigenvalue of $A$ and $\dim(\ker (A - \lambda I)) = 1$.($\lambda$ has geometric multiplcity one) why is $\text{adj}(A - \lambda I) \ne 0$?
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39 views

I have to show $\operatorname{rank} X$ and $\operatorname{rank} T$ are at most one. [closed]

Assume that $$B=\begin{pmatrix} X & Y \\ Z & T \end{pmatrix}$$ is an idempotent matrix, where $Y$ is an $r \times (n-r)$ matrix. I know $\operatorname{rank} B=1$. I have to show ...
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111 views

Positive semi-definite matrix and exponent of its entries

Let $(a_{ij})_{i,j=1,...,n}$ be a symmetric positive semi-definite matrix with real entries. Is the matrix $(\exp(a_{ij}))_{i,j=1,...,n}$ a positive semi-definite?
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$A$ and $B$ are similar.why there are $X$ and $Y$ s.t $A=XY$ and $B=YX$

let $A,B \in {M_n}$, $A$ and $B$ are similar.why there are $X$ and $Y$ s.t $A=XY$ and $B=YX$ . (NOTE: $X$ or $Y$ is nonsingular)
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1answer
115 views

Do scalars commute across matrices?

Do scalars commute across matrices? $A,B,C$ are matrices that work together, lets just assume they are all $n\times n$, and $a$ is a scalar. E.g. does $aABC=AaBC=ABaC=ABCa$, I imagine this is the ...
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1answer
48 views

The trace of inner product

I am reading a matrix algebra textbook. It introduces an equation $(x-a)^TA(x-a)=tr(Ax_cx_c^T)+n(a-\bar{x})^2tr(A)$ where $ x_c = x - \bar{x}$. Is this equation really right? I failed to prove it. ...
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1answer
59 views

Determining whether the line $\frac{x-2}{2} =\frac{y+2}{-4} =\frac{z+3}{9}$ contains the point $(3,-4,1)$

How can I determine whether the line $\dfrac{x-2}{2} =\dfrac{y+2}{-4} =\dfrac{z+3}{9}$ contains the point $(3,-4,1)$?
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1answer
66 views

How to find a basis of eigenvectors?

Let $\mathfrak{sl}_2$ be the vector space of $2\times 2$ traceless matrices. Let $A\in \mathfrak{sl}_2$ be a diagonal matrix. Define a linear operator $\phi_A: sl_2\to sl_2$: $$\phi_A(X)=AX-XA$$ I ...
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38 views

How to find a partial derivative with respect to a matrix?

Let we have a $2\times2$ matrix $A=\begin{bmatrix}a_1&a_2\\a_3&a_4\end{bmatrix}$, a $1\times2$ matrix $C$, and a $2\times1$ matrix $X$. How can we calculate derivative of $CAX$ with respect to ...
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35 views

How to write this to a linear programming problem?

A procedure of animal feed makes two food products: F1 and F2. The products contain three major ingredients: M1, M2, and M3. Each ton of F1 requires 200 pounds of M1, 100 pounds of M2, and 100 pounds ...
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1answer
93 views

Theorem on characteristic polynomials and minimal polynomials.

Let $A$ be an $n\times n$ matrix. Let $q_A(t)$ and $p_A(t)$ represent the minimal and characteristic polynomial respectively. Then, the following are equivalent: (a) ...