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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Orthographic projection in euclidean space

Let $E$ be a euclidean space with an inner product given by $$B =\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 2 \end{array} \right) $$ in a basis ...
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1answer
38 views

Square matrices A and B commute if and only if they share the same generalized eigenspace.

I found a couple of proofs for this theorem but only for the case when A and B are diagonalizable, thus the eigenspace that they share is not the generalized one. Im looking for the proof (or ...
1
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3answers
30 views

Basis of a vector space is a maximal linearly-independent set?

If $V$ is a vector space of finite dimension over $F$, then a basis of $V$ is a maximal, linearly independent set in $V$. Is this conjecture true? If so, how to prove it?
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1answer
20 views

Linear Algebra - Weighted Inner Product of Polynomials [on hold]

Given the weighted inner product $\langle f,g\rangle = \int^1_{-1}f(x)g(x)x^2dx$ How do you find an orthogonal basis of the space $\Bbb P^1$ of polynomials of degree $\le$ 1. And how do you find the ...
2
votes
2answers
55 views

derivative of a symmetric bilinear form (quadratic form version)

Let $A=A^T\in \mathbb R^{k\times k}$ be a nonzero symmetric matrix and define $F:\mathbb R^k\to\mathbb R$ by $$f(x):=x^TAx$$ Then why $df(x)\xi=2x^TA\xi$ for $x,\xi\in\mathbb R^k$?
3
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2answers
38 views

Is operator norm invariant under multiplication with orthonormal matrix

Let $A$ be an $r \times r$ real matrix and $Q$ be a $n\times r$ matrix whose columns are orthonormal. I know that $||QA||_2 = ||A||_2$ because $$||QA||_2 = \max_{||x||_2=1} ||QAx||_2\,= ...
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1answer
45 views

Linear Algebra - backward substitution

Is it faster to do back substitution that multiplying A by a vector?
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1answer
23 views

Transformation of coordinate axis to make matrix diagonal

Consider the matrix $$ A= \begin{bmatrix}1/8 & \frac{-5}{8\sqrt{3}} \\ \frac{-5}{8\sqrt{3}} & 11/8 \end{bmatrix} $$ which of the following transformations of the coordinate ...
0
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1answer
38 views

Square root of this $2x2$ matrix

I am to find $R$ of $A$ from the following formula: $R=S\sqrt{\Lambda }S^{-1}$. I however am getting the wrong answer (not the same as the textbook answer). The answer in the textbook is: ...
0
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1answer
23 views

How to define a transition matrix mathematically?

I'm writing my master thesis. Given the adjacency matrix of a graph, I need to define the transition matrix formally. I'm not able to figure out how to define it in mathematical notation. Can you help ...
0
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2answers
20 views

Finding a basis and dimension for subspaces of $R^{4}$

I have an homework question I can't understand how to solve. Let $U$ and $W$ be subspaces of $\mathbb{R}^{4}$ such that $$\dim U>\dim W,$$ $$U\cap ...
0
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1answer
18 views

Linear Algebra - Inner Products, Functions, and Closet Polynomial

This is the question: Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt ...
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1answer
24 views

$\left\Vert J(x)^{-1}\right\Vert<2\left\Vert J(x^*)^{-1}\right\Vert. $?

Could you please help me to prove this theorem: Suppose $J:{\bf {\rm R}}^m\rightarrow{\bf {\rm R}}^{n\times n}$ is a continuous matrix-valued function. If J(x*) is nonsingular, then there exists ...
3
votes
1answer
71 views

Why is this change of basis useful?

In my textbook there is a theorem which states Let $A$ be a real $2\times 2$ matrix with complex eigenvalues $\lambda =a\pm bi$ (where $(b\ne 0)$. If $\mathbf x$ is an eigenvector of $A$ ...
1
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1answer
17 views

How to derive one formula of projection from another

The projection onto the subspace (hyperplane) $$H:=\{x\in \mathbb{R}^n:\langle a,x\rangle=0\}$$ is given by $$P_{H}(x)=x-\frac{\langle a,x\rangle}{\|a\|^2}a. \tag 1$$ and also the projection onto ...
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2answers
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How to calculate $R=S\sqrt{\Lambda}S^{-1}$

When I think about it, I've never actually calculated this before, $R=S\sqrt{\Lambda}S^{-1}$. $S=\begin{bmatrix} 1 & 1\\ -1 & 1 \end{bmatrix}$ $S^{-1}=\begin{bmatrix} 1 & -1\\ 1 & ...
1
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1answer
38 views

Determinant of nXn matrix

I know this was already asked before here: Q: The determinant of a NxN matrix? But I still did not manage to solve this with the method he suggested. I tried adding all the columns to the first one ...
0
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0answers
28 views

Understanding change-of-basis and linear operators

First of all , apologies in advance as this isn't so much as a question, but more check of my understanding. Suppose I have an $n$-dimensional vector space $V$ and a given basis $\mathfrak{B}= ...
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0answers
12 views

Location and perturbation of eigenvalues

This is a problem from Horn and Johnson's Matrix Analysis. I'm having trouble showing the bolded parts in the following paragraphs. In fact, I don't really understand what the sentences mean. I would ...
2
votes
2answers
48 views

Square root of these $2\times2$ matrices

I am to find the matrix square root of $A$ from the following formula: $R=S^{-1}\sqrt{\Lambda S}$ and explain why there is no real matrix square root of $B$. I am stuck on A as the following ...
2
votes
1answer
28 views

Unit close disc to prove a matrix algebra identity?

I need to prove that every $3 \times 3$ matrix with real positive entries has one eigenvector with a positive eigenvalue. Now, how do I prove this using the fact that the set $B=\{x=(x_1,x_2,x_3)\in ...
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1answer
16 views

Product of $L_2$ norm of vectors

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ? where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.
2
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1answer
28 views

$A_i \sim B_i \implies \text{Diag}(A_1 \ldots A_n) \sim \text{Diag}(B_1\ldots B_n) $ [closed]

How do I prove that: $A_i \sim B_i \implies \text{Diag}(A_1 \ldots A_n) \sim \text{Diag}(B_1\ldots B_n) $ Notation: $A\sim B$ meaning is $A$ is similar to $B$. Also, $A_i, B_i$ are square matrices ...
3
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1answer
48 views

How do we get from one fromula to another?

My question is about the projection. The projection onto a hyperplane defined by $$H:=\{x\in \mathbb{R}^n:\langle a,x\rangle=b\}$$ is defined by $$P_{H}(x)=x-\frac{\langle ...
2
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0answers
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Column and row vectors (spinors) in Landau-Lifshitz vol.IV Theoretical Physics

I am getting confused by the notation the authors of this book since they define: $$ \bar{\psi}\equiv \psi^\ast \gamma^0 $$ where (I suppose) $^\ast$ means complex conjugate and $\gamma^0$ is one of ...
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1answer
27 views

Eigenvalue inequalities for Hermitian matrices

This is a problem from Horn and Johnson's Matrix Analysis. I've tried to follow the problem but I can't find a way to lead to the conclusion the problem is suggesting. Any solutions, hints, or ...
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votes
2answers
66 views

Linear maps for $\Bbb{R}^n$ to $\Bbb{R}^m$?

This question is related to: What is $\Bbb{R}^n$? The basis of a matrix representation I am still confused about the topics in these questions and am going to ask another question that will ...
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2answers
166 views

If A is normal, then the nullspace of A is the nullspace of A*

Suppose $A$ is a normal matrix. Prove that $x$ is in the nullspace of $A$ if and only if $x$ is in the nullspace of $A^{*}$. This isn't a homework problem. It was on a test I took recently, and I'd ...
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3answers
41 views

Linear Algebra: Polynomials Basis

Consider the polynomials $$p_1(x) = 1 - x^2,\;p_2(x) = x(1-x),\;p_3(x) = x(1+x)$$ Show that $\{p_1(x),\,p_2(x),\,p_3(x)\}$ is a basis for $\Bbb P^2$. My question is how do you even go about proving ...
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3answers
35 views

How to prove that the dimension of a hyperplane is n-1

The hyperplane $H$ defined by $$H:=\{x\in\mathbb {R}^n:a^Tx=b\}$$ is the set that has dimension $n-1$, my question is why or how can we prove that its dimension is $n-1$? Thank you to every one who ...
6
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0answers
47 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.
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0answers
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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 ...
0
votes
3answers
79 views

Why does the determinant $D$, have to be $0$ for equation to have a solution?

Suppose $2\times2$ equation: $$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} $$ We can make determinants: ...
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 ...
2
votes
3answers
49 views

If $v_1,…,v_r$ are the eigenvectors that correspond to distinct eigenvalues, then they are linearly independent.

Prove: If $v_1,...,v_r$ are the eigenvectors that correspond to distinct eigenvalues $\lambda_1, ...,\lambda_r$ of an $n \times n$ matrix $A$, then the set $\{v_1,...,v_r\}$ is linearly ...
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, ...
1
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1answer
22 views

linear function of a convex set

Suppose K $\subset \mathbb{R^2}$ is a convex set. If $x\in\partial K$, which means the boundary of K, show there is a linear function $l_x:\mathbb{R^2}\rightarrow\mathbb{R}$ such that $l_x(x)\geq ...
0
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1answer
32 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
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1answer
50 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 ...
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0answers
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case deletion formula restricted least square estimator [closed]

hi any one please help me to find out a case deletion formula for restricted least square estimator? $$ \hat\beta = (X' X)^{-1} X'y-(X' X)^{-1} R' [R' (X' X)^{-1} R]^{-1} R(X' X)^{-1} X'y $$ i need a ...
5
votes
4answers
72 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 ...
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 $B\in \Bbb R^{n\times h}$ ...
0
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1answer
17 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 ...
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3answers
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Let V be the set of real numbers. Regard V as a vector space over the field of rational numbers, with the usual operations [closed]

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.
3
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2answers
30 views

Common formula for two sets of linear data

I've got two sets of data, both of which create a roughly linear line. I developed each set with a value of K, and I collected the data. ...
2
votes
3answers
49 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 ...
2
votes
1answer
29 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
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 ...