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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6
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45 views

A polynomial that annihilates two other

While studying, I found the following problem: Let $f, g \in F[t]$. Prove that $\exists p \in F[x, y], p \neq 0 : p(f(t), g(t)) = 0$ I'd thank any hints that point me in the right direction.
0
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0answers
15 views

Derivative Nullity for nonpolynomial spaces

One thing has been bothering me about derivatives, it's easy to explain nullity of a polynomial, since a term that is constant after n many derivatives will become zero at n+1 many derivatives. How ...
0
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0answers
22 views

A problem on Gersgorin cirle passing through the eigenvalue of an absolute matrix

I'm having trouble solving the following problem. I think I need to show that the matrix $D^{-1}|A|D$ has property SC, but I can't come up with a way to show it. I would really appreciate any ...
2
votes
1answer
45 views

Proving that quadratic form is convex in (vector, matrix) arguments

I'm studying with the quadratic form $$ F( (x,Q) ) = \langle x,Q^{-1}x\rangle $$ considered over $\mathbb{R}^n\times\mathbb{R}^{n\times n}_+$, where $\mathbb{R}^{n\times n}_+$ is the set of all ...
2
votes
1answer
25 views

Dimension of Null and zero singular values

Suppose $T\in L(V)$. Prove that $\dim(\operatorname{null}(T))$ is equal to the number of zero singular values of T. Proof. Suppose $T\in L(V)$. By Singular-Value Decomposition, $T$ has singular ...
0
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1answer
15 views

A problem about a theorem on irreducible matrix

I'm stuck on a problem where I need to find a counterexample. I'm not sure how to come up with a reducible matrix to show that it doesn't satisfy the result of the following corollary. Any solutions, ...
3
votes
1answer
316 views

Covariance- v. correlation-matrix based PCA

In principal component analysis (PCA), one can choose either the covariance matrix or the correlation matrix to find the components. These give different results because, I suspect, the eigenvectors ...
5
votes
1answer
153 views

Typo in the book or am I going crazy?

I am reading about integral bases from Frazer Jarvis' "Algebraic Number Theory", but my question is really about elementary linear algebra. In page 49, author claims the following: I don't think ...
0
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1answer
31 views

Non-simplifiable permutation matrices

The permutation matrices for 2 and 3 dimensions look like this: 2-dimensional: $$\quad M_1^{2d}=\left(\begin{matrix}1 &0\\0 &1\end{matrix}\right), M_2^{2d}=\left(\begin{matrix}0 &1\\1 ...
0
votes
1answer
37 views

What values must $\alpha$ be so that $F$ is an isomorphic linear transformation? (Bijective)

Let $F:P_2\to P_2$ where $P_2$ is a polynomial vector space with max grade of 2. $$[F]_B= \begin{pmatrix} \alpha & -1 & -1 \\ -6 & \alpha +1 & 0 \\ ...
2
votes
1answer
28 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
2
votes
1answer
48 views

Is there an error in my matrix proofs (Also: potato quality jpeg errors present)

Disclaimer: The jpg quality of the problem is terrible, ALL SUPERSCRIPT IN BLOCKQUOTES CAN BE INNACURATE. $A\in \Bbb R^{n\times n} $ is symmetric $B\in \Bbb R^{n\times h}$ ...
5
votes
4answers
70 views

Whether a $2 \times 2$ matrix of rank $1$ has a zero eigenvalue

"Does $A = \begin{bmatrix}1&2\\2&4\end{bmatrix}$ have a zero eigenvalue?" Well, it would be a funny question to ask if the asker didn't state that he wants us to explain without computing the ...
0
votes
1answer
16 views

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$ $\lt C_k'$, for at least one value of $k$

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$$\gt C_k'$, for at least one value of $k=1,\dots, n$, where $C_k'$ denotes $A$'s deleted absolute column sums ($a_{kk}$ is ...
-3
votes
3answers
45 views

Let V be the set of real numbers. Regard V as a vector space over the field of rational numbers, with the usual operations [on hold]

Let V be the set of real numbers. Regard V as a vector space over the field of rational numbers, with the usual operations. Prove that this vector space is not finite-dimensional.
0
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1answer
22 views

Find bases for orthogonal complement $S^\perp$ for the subspace $S$

I'm having a tough time understanding the textbook on how to answer this question? I'm not too sure what to do? Any help will be appreciated. $$ S=\operatorname{span}\left[ \begin{pmatrix} 1 \\ -3 ...
3
votes
2answers
55 views

Find Jordan form of a $3\times 3$ matrix

$$\left( \begin{array}{ccc} 0 & 1 & 2 \\ -5 &-3 & -7 \\ 1 & 0 & 0 \end{array} \right) $$ I figured out the eigenvalues are all -1 from the characteristic polynomial, but I'm ...
2
votes
1answer
18 views

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. If $A$ is real, show that every eigenvalue of $A$ is real.

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. (a) If $A$ is real, show that every eigenvalue of $A$ is real. (b) If $A \in M_n$ has real main diagonal entries and its ...
2
votes
3answers
47 views

$V=V_1 \oplus V_2 = V_1 \oplus V_3$ does not mean $V_2 = V_3$

My professor told me that $V=V_1 \oplus V_2 = V_1 \oplus V_3$ does not imply $V_2 = V_3$. I would like to see an example of this claim. What conditions do I need to have "if $V=V_1 \oplus V_2 = V_1 ...
0
votes
1answer
30 views

How to demonstrate a set is a real vector space (set governed by nonstandard operations)

I am really not that familiar with questions that ask you to work with a operation vector space, even less with the English terms for it. I am... quite lost. How would you prove that it is a real ...
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0answers
45 views

Proving the existence of an invertible square matrix

Assume $A$ is a square matrix with real values. Show that there exist an invertible square matrix $B$ such that matrix $B^{-1}AB$ is block upper triangular with diagonal blocks either of size ...
2
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1answer
15 views

Show that the intersection taken over the Gersgorin discs of all similar matrices of $A$ $=$ $\sigma (A)$

Show that $\bigcap_S G(S^{-1}AS)$ $=$ $\sigma (A)$; the intersection is taken over all nonsingular $S$, and $\sigma (A)$ is the spectrum of $A$. I'm lost as how to even begin to prove this fact. Any ...
3
votes
3answers
80 views

Let $T:V\to W$ be a linear transformation. If $\dim V> \dim W$ then $T$ is not injective. True or false?

I think it's true, I just did this demo, please can you help me if I'm missing something or doing it wrong. Thanks. Let $T\colon V \to W$ a linear transformation. If $\dim V > \dim W$, then $T$ ...
1
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3answers
37 views

orthogonal and special orthogonal group of dimension $2$, group of isometries of $S_1$, $\mathbb{R}^2$ [on hold]

In my abstract algebra class, my teacher gave us this problem as to help review for the final. Unfortunately, I am not very well versed with linear algebra so I don't understand all that well what ...
2
votes
1answer
45 views

Exercise on projections

Let $f\in\mathcal{L}(E)$, the set of linear maps $E\rightarrow E$, where $E$ is a vector space of dimension $n$. How can I show that there exists two projections $p,q$ (i.e., maps $p$ such that $p^2 ...
1
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1answer
18 views

Prove: the sum of simultaneously diagonalizable transformations is diagonalizable

Let $T, S$, linear transformations which are simultaneously diagonalizable. Prove that $T+S$ is diagonalizable. I need to rely on the the definition: $T,S$ are called simultaneously ...
2
votes
1answer
18 views

Relationship between similarity and having the same minimal polynomial

Let $A$, $B$ $\in M_3$ be nilpotent, where $M_3$ is the set of all complex 3by3 matrices. Show that $A$ and $B$ are similar if and only if $A$ and $B$ have the same minimal polynomial. Is this true in ...
1
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2answers
22 views

Proof that the kernel of an endomorphism to the power $n$ is a subset of the kernel of the endomorphism to the power $n+1$

I am expected to know how to prove the following but I can't seem to draw it out. Knowing that V is a Vector Space$$ T:V\to V $$ Prove the following $$ Ker(T^n)\subseteq Ker(T^{n+1}) $$ How ...
1
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1answer
19 views

Applying Gram-Schmidt process to a set of vectors to find first three polynomials orthogonal with respect to inner product

$.$ $\langle f, g \rangle = \displaystyle\int_{-1}^{1} f(x)g(x)dx$ Apply Gram-Shmidt process to the set of vectors $:$ {1, x, $x^2$, ...} to find the first three polynomials orthogonal with respect ...
2
votes
2answers
41 views

$A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive.

Let $A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive. Then which of the following statements is always false? A. ...
4
votes
2answers
32 views

Theorem with seemingly reduntant part

I encountered the following theorem in a linear algebra book: For any vectors $u, v$ in $R^n$ and any scalar $k$ in $R$: $u . u \geq 0$, and $u . u = 0 \iff u = 0$ I found the theorem in almost the ...
1
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0answers
19 views

interpertation of the determinant of an X'X product matrix (D-optimal design application)

As the title suggests, I have been looking into an application of D-optimal design. I read this thread What does it mean to have a determinant equal to zero? and found some of the answers ...
0
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1answer
30 views

Finding eigenvvalue and eigenspace

I am given a matrix $A= \bigg({} \matrix{10 & 7 \\-14 &-11} \bigg{)}$ and eigenvalue $3$. My elite mission is to find the treacherous basis for the eigenspace. I used the $(A -eI)=v$ where ...
2
votes
2answers
27 views

Question about diagonalization and projections

Let a finite dimensional vector space $V$ above $\mathbb{F}$. Let $T:V\to V$ a diagonlizable transformation. We denote $a_1 \ldots a_r$ the $r$ different eigenvalues of $T$. By diagonalization, we ...
1
vote
1answer
21 views

Using trigonometric identity to compute an inner product through an integral to form an orthogonal sequence of functions.

Consider the inner product space: $(C(0,L),\langle \cdot,\cdot \rangle),$ where: $\langle f,g \rangle = \displaystyle\int_0^L f(x)g(x)\, dx$. Use the trigonometric identity: $\sin(u)\sin(v) = ...
0
votes
2answers
47 views

Solution to a system of linear equations with an unknown matrix product

Consider the system of equations $$ Xy=Ab $$ where $X$ and $A$ are $m \times m$ invertible matrices and $y$ and $b$ are $m \times n$ matrices. The matrices $X$ and $y$ are unknown and the matrices $A$ ...
0
votes
2answers
23 views

Can I just use the following notation when proving a set is a vector space?

If given all functions of form $$f(x) = a + b \cos(x) + c \sin(x),$$ where $a,b,c$ are real numbers, would it be sufficient to use the notation "$f(x)$," when proving that the axioms hold and that ...
2
votes
1answer
36 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
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1answer
15 views

Projection of vectors

Compute $:$ $proj_\vec y (\vec x)$ $\vec{x}_1=\begin{bmatrix} 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}, \vec{y}_2 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix}$ Since the projection would be $:$ $(-2/0) * ...
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votes
0answers
41 views

Thereom on Bases of Finite Vector Spaces

Please produce a proof of the following... Theorem: If $B_1$ and $B_2$ are two bases of a finite vector space $V$, then for all $\vec x\in B_1$, there exists $\vec y\in B_2$ such that $(B_1 - \{\vec ...
2
votes
1answer
326 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
3
votes
2answers
46 views

This answer is confusing $4\times 4$ eigenvalue calculation

Question: Find the rank and the four eigenvalues of the following matrix: $\begin{bmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & ...
1
vote
1answer
25 views

Solving linear differential equations system

Upon trying to solve this particular system , I've encountered a few problems. $$ y'=5y+4z $$ $$z'=-4y-3z$$ After solving for eigenvalues the quadratic yielded a double root at $\lambda=1$ . But I ...
1
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1answer
28 views

Quadratic form - vector/matrix

I have two very simple (stupid) questions about quadratic forms. Having any matrices $A,B$ and vectors $x,y$ (real/complex, singular/regular, rectangular, infinite size, etc.) with appropriate size ...
4
votes
2answers
54 views

Some questions about the Eigenvalues of this $4\times 4$ matrix

\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \end{bmatrix} The rank is $1$ as there is only $1$ linearly ...
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2answers
35 views

How to prove that a set is a vector space

How does one, formally, prove that something is a vector space. Take the following classic example: set of all functions of form $f(x) = á_0 + a_1x + a_2x^2$, where $a_i \in \mathbb{R}$. Prove ...
2
votes
1answer
24 views

Polynomial functions/basis

If I suppose $R \subset F$ and have polynomial functions $p_{k,j} : F \to F$ by $p_{1,0}(x)=(x-2)^3$ $p_{2,0}(x)=(x-1)$ $p_{2,1}(x)=(x-1)(x-2)$ $p_{2,2}(x)=(x-1)(x-2)^2$ and the polynomial ...
1
vote
2answers
21 views

Question in regards to definition: finite dimensional

Do we denote a vector space as finite dimensional IF it has a basis, or do we say that it is finite dimensional if it's associated through an isomorphic transformation with a "number space", ie. ...
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vote
2answers
25 views

Inner prodcut space for complex numbers including complex conjugation

$..$ Consider inner product space : $(C, \langle \cdot,\cdot\rangle)$: where for complex numbers $..$ $\langle z_1, z_2 \rangle = \sqrt(z_1 *\overline{z_2}$) Computing $..$ $\langle 2-3i, 2-3i ...
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votes
0answers
25 views

Null space, column space and row space be a line [duplicate]

For a 4x3 matrix can the nullspace, the column space and row space all be a line through the origin? For a 2x4 matrix can the nullspace, the column space and row space all be a plane through the ...