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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Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$.

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$. $A$, $B$, and $C$ are square matrices of the same size. $\hat{c}_j$ is the $j$th column of $C$, $\hat{a}_k$ are the columns of $A$, ...
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0answers
32 views

1-How my profesor reach this solution? 2-How can I use eigenvalues to compute betas?… if there is any way

this time I quite don't undertand how the profesor avoid using matrix algebra to solve this exercise. Statement: Below you can see a scatter plot of the data with the three regression lines ...
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2answers
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Adjoint of linear transformation $T: \mathbb{M_n(C)} \rightarrow \mathbb{M_n(C)}$

Let V = ${M_n(\mathbb C)}$ with inner product $\langle A, B\rangle = \text{tr}\,(B^*A)$, $A, B \in V$. Let $M \in {M_n(\mathbb C)}$, Define $T: V \rightarrow V$ by $T(A) = MA$. What is adjoint of ...
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1answer
15 views

determinant inequality for Hermitian matrix

$A \in \mathbb{C}^{M \times M}$ is a positive semidefinite matrix with all diagonal entries being $1$. and the vector $\mathbf{y} \in \mathbb{C}^{M}$ has entries $|y_{i}| < 1$. Prove that $$2 ...
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Property of group of $k \times k$ orthogonal matrices

Does the group of $k \times k$ orthogonal matrices lie on an $(k^2 - 1)$-sphere? If so, of what radius? If not, does it lie on some sort of other object?
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3answers
28 views

Eigenvalues of a 2 by 2 matrix

How do I show that if the eigenvalues of a 2x2 matrix A is 0 and 1, then $A^2=A$. I know that the if $A^2=A$, then the eigenvalues of A are 0 and 1. But I have no idea how to prove the this problem.
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33 views

Find a basis for subspace $W$

Let $ W=\{(a,b,c,d,e)\in \mathbb R^5:a=3c,2b-4d+5e=0\}$ I am looking for a basis for this subspace of $\mathbb R^5$. I do not remember the way to do this, can someone give me a hint?
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1answer
37 views

Vector space endomorphisms in $\mathbb{R}[x]$ commuting with $E:f\mapsto f+f'$

I am wondering if every vector space endomorphism in $\mathbb{R}[x]$ that commutes with $E:f\rightarrow f+f'$ is invertible. (denoting $f'$ the derivative of $f$) To begin with, $E$ is invertible ...
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27 views

For what positive integers $n$ and m are $\sin (nx)$ and $\sin$ $(mx)$ orthogonal over $0 ≤ x ≤ 2π?$

The question is: For what positive integers $n$ and m are $\sin (nx)$ and $\sin$ $(mx)$ orthogonal over $0 ≤ x ≤ 2π?$. Let $f(x) = \sin(nx) $ and $g(x)= \sin(mx)$ Using this formula ...
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Linear and Gondran-Minoux independence in (Max,+) Algebra

I'm reading Peter Butkovic's monograph Max-Linear Systems: Theory and Algorithms. In Chapter 6, linear independence and Gondran-Minoux independence are introduced. A set of vectors $\{a_1, a_2, \dots ...
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Properties of Determinants - True or False [on hold]

Can you help me answer these true or false questions for an n x n matrix A? I think that 3 and 10 are actually false Picture of the problem The determinant of a lower-triangular matrix A is the sum ...
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25 views

Divisibility of dimension by matrix equations.

Let ${\bf M}$ and ${\bf N}$ be two real $k\times k$ matrices such that ${\bf M}^2+{\bf N}^2={\bf M}{\bf N}$. Show that if $\det\left({\bf N}{\bf M}-{\bf M}{\bf N}\right)\neq 0$, then $3\mid k$.
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Related to inverse function theorem

We're using a book that has some really bad typos and I just want to make sure this exercise from it doesn't contain a serious one. This is the problem exactly as it's written in the text: Let $U ...
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2answers
30 views

Basis of a 3x3 eigenspace

I'm currently in the middle of a question where I'm given a 3x3 matrix: $$\left(\begin{array}{rrr} 3 & 0 & 0\\ -2 & 7 & 0\\ 4 & 8 & 1 \end{array}\right).$$ and have been ...
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1answer
16 views

Diagonal block matrices of a positive definite block matrix

Let $R=\begin{bmatrix} R^{11} & R^{12} & R^{13} \\ R^{21} & R^{22} & R^{23}\\ R^{31} & R^{32} & R^{33} \end{bmatrix}$ be a symmetric positive definite matrix where $R^{ii}$, ...
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29 views

If $v^T A^{-1} u = -1$, then the matrix $A + uv^T$ isn't invertible

Let $A \in M_n(\mathbb{R})$ be an invertible matrix and $u,v \in \mathbb{R^n}$. By the Sherman Morrison formula, we know that if $v^TA^{-1}u \neq -1$ then $(A + uv^T)^{-1}$ exists. I want to prove ...
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29 views

For $T:\mathbb{R}^6 \to \mathbb{R}^6$ and $T^5 \neq 0, \; T^6 = 0,$ prove there exists no $S$ such that $S^2 = T.$

Let $T:V \to V$ be a linear operator such that $T^5 \neq 0,$ but $T^6 = 0.$ Suppose $V$ is isomorphic to $\mathbb{R}^6.$ Prove that there does not exist an $S:V \to V$ such that $S^2 = T.$ Does the ...
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How do I determine $\min \left \| \vec{v}-\vec{u} \right \|_2$ for $\vec{u}\in U$?

Let $U=\lambda ((1, 0, 1, 0)^T,(1, 1, 0, 1)^T,(1, -1, 1; 0)^T$ is a subspace of $\mathbb{R}^4$. Determine for $\vec{v} = (1, 1, 1, 1)^T$ the vector $\vec{u}\in U$ minimal with $\left \| ...
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18 views

Does Linear Transformation transforms orthonormal bases of symmetric matrix into orthogonal vectors?

How would you show it? I went all over the book: looked at Linear Transformation definition again, looked at orthogonality of bases for symmetric matrices. I got to the point: Since L is linear ...
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1answer
63 views

prove that $TS = ST$

Let $V$ be a finite-dimensional vector space over $F$ with $\dim(V) = n$. Let $ T:V \rightarrow V$ be a linear transformation. Assuming that $T$ has $n$ different eigenvalues. prove that :$$ TS = ST ...
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1answer
14 views

Finding the inverse of a map given in vector form.

The question asks me to find the inverse map $ \mathbf\Phi^{-1} $, of: $$ \mathbf{\Phi}(\mathbf{x})= \mathbf{n\lor(x \lor n)} + \alpha\mathbf{(n \cdotp x)n} $$ for $\alpha$ such that the inverse ...
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3answers
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How to chart points on a stretched line

I'm somewhat math literate but need help on a formula. Lets say I have a straight line that starts at 8500 and ends at 11,700 with seven points in between at irregular intervals between each other. ...
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2answers
29 views

The image of the transpose of $A^T$ is the orthogonal complement of its kernel

Suppose $V$ is a finite dimensional vector space over $\mathbb{K}$. Let A be a linear map. I am trying to prove that $$ImA^T=(KerA)^{\perp}$$ I know one direction: $ImA^T \subset (KerA)^{\perp}$ but I ...
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6answers
37 views

Determine whether $w$ is in the $Span\{v_1, v_2, v_3\}$

my question is how to determine whether a vector $w$ is in the $span\{v_1, v_2, v_3\}$. In this case: $w = \begin{bmatrix} 9 \\ 6 \\ 1 \\ 9 \\ ...
2
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2answers
65 views

When this matrix is diagonalizable?

When this matrix is diagonalizable? ($a_i \in \mathbb{R}$) $$ \begin{pmatrix} &&&a_1\\ &&a_2&\\ &\ddots&&\\ a_n&&&\\ \end{pmatrix} $$ I think I should ...
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0answers
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If $(AA^\dagger+A^\dagger A)\vec v\propto \vec v$ is $A A^\dagger \vec v\propto \vec v$?

Let us say we have a matrix $A$, such that the vector $\vec v$ is an eigenvector of: $$A A^\dagger +A^\dagger A$$ i.e.: $$(A A^\dagger +A^\dagger A)\vec v=\lambda \vec v$$ Is it possible to say that: ...
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Linear transformation $R^3 \to R^2$

Hi just wondering if anyone can give a hand with this question Suppose $T : R^3 → R^2$ is defined by $T(x, y, z) = (x − y + z, z − 2)$, for $(x, y, z) ∈ R^3$ . Is T a linear transformation? Justify ...
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1answer
48 views

How can I compute the angle between $f(x)=3x^2+2$ and $g(x)=2x+3$?

Compute the angle between the vectors: $$f(x)=3x^2+2 \\g(x)=2x+3$$ Using the standard dot product: $a \cdot b=\lvert a \rvert \lvert b\rvert \cos{\alpha} \iff \alpha=\cos^{-1}\left(\frac{a \cdot ...
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1answer
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Finding the basis of a set

I'm having a bit of trouble with Linear algebra and just trying to cram for exams. I'm currently looking at exam questions and came across these and I'm stumped any help would be really appreciated. ...
3
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1answer
62 views

Result about Matrices of form $B(AB)^{-1}A$

I am trying to prove the following result. So far my only idea was to try using the formula for inversion of block matrices, but that did not get me very far. Any help will be much appreciated. ...
3
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3answers
61 views

Proofs in linear algebra

I'm pretty awful at proving linear algebra proofs, I just don't understand how you know what to do or where the information comes from. I have some sample questions below of what I mean, I have no ...
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1answer
9 views

solving Work distribution related problem easily

If Mar, Rox, & Haz writes 9, 4 & 10 hours respectively, then 7 stories are made. If Mar writes 3 hours, Rox writes 8 hours, then 5 stories are made. If Rox writes 6 hours, Haz writes 5 hours, ...
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1answer
44 views

How to prove that $\det A=|\lambda|^n$

Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$ .Let $V_0$ be the subspace of $V$ which forms the image of $V$ under $A$. Let $k=\dim V_0<n$ and suppose ...
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1answer
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Consider the following change of basis matrix [on hold]

Consider the $n$ dimensional vector space $\mathbb{V}$ over the reals with two basis $B$ and $B'$. Show that the transition matrix maps coordinate vectors of the basis $B'$ to coordinate vectors in ...
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14 views

Connection between power iterations and QR Algorithm

I am seeking an intuitive understanding of why the QR Algorithm solves the symmetric eigenvalue problem. In class, and also in Golub and Van Loan, it has been suggested that there is somehow deep ...
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Do $A$ and $B$ have the same eigenvalues?

There is a property that I've been told by my quantum physics teacher that states: Being $A$ and $B$ Hermitian, if $AB-BA=0$ then $A$ and $B$ have the same eigenvalues. I do not even know how to ...
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2answers
47 views

Matrix in linear algebra

would you please help me solve this question? $A$, $B$, $C$ are matrices. if $AB = AC = I$ and $BA = CA = I$, show that $B=C$ I think we can't use inverse matrix for this problem. A,B,C are in the ...
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2answers
61 views

If $A$ is normal and upper triangular then it is diagonal

Let $A$ be a normal matrix in Mat$_{n\times n}(\mathbb C)$, if $A$ is upper triangular then it is diagonal (Normal means $AA^*=A^*A$, where $A^*$ is the conjugate transpose of $A$) If I consider ...
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With these two equations, how do I show that either a,b,c,d must be negative, if v is not 0?

If I have the equations $$ad-bc = u^2 +v^2$$ $$a+d = 2u$$ and I want $a, b, c, d \ge 0$, then how I can show that this is impossible, if $v \ne 0$? I.e., if $v \ne 0$, then one of $a,b,c,d$ must ...
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For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
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Property of Gauß Elimination

I have the following problem: Suppose $A\in\mathbb{R}^{m,n}$ has the property that after applying the Gauß Elimination we obtain a row-echelon form but without using changes of rows.The claim should ...
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1answer
51 views

Square root of matrix that is a square of skew-symmetric matrix

Let's suppose we have a matrix $A$ (dimension $3\times 3$) which is the square of some skew-symmetric matrix $S$ i.e. $A=S^2$. How to obtain from $A$ its skew-symmetric square root $S$?
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Matrices Eqvialence Relation

How can I prove that $A\mathcal{R}B$ is an equivalence relation if there exists an invertible matrix $C$ such that $B = CA$? I know there there is a reflexive, symmetric, and transitive steps. ...
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59 views

For what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$

For what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$ I tried to solve this question by differentiating and making it equal to $0$ and solving for $x$ and i got $-10$ as an ...
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From Cosine formula between two vectors to Schwarz inequality and Triangle inequality?

I'm studying linear algebra. I'm a beginner in this subject. The book says: Cosine formula of two vectors $$\frac{v \cdot w}{||v|| ||w||} = cos\theta $$ and it says, since $|cos\theta|$ never exceeds ...
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48 views

Prove that the product of two invertible matrices also invertible

I am working on a homework problem, but I am lacking some understanding. Here is the problem: Let $A$ and $B$ be invertible $n \times n$ matrices with $\det(A) = 3$ and $\det(B) = 4$. I know that ...
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21 views

linear algebra problem 12 [on hold]

In rotation in three dimensions, the square of length of any position vector does not change. Using the matrix representation of such transformation, show that three independent angles are sufficient ...
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35 views

Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
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28 views

Positivity of the last component of non negative least squares based on active set method

I have followed the instructions given in Lawson and Hanson book for non-negative least squares using active set method. I am having a trouble in justifying one of the statements they have made about ...
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20 views

Finding bases of subspaces of a linear map

So i'd like to find out a basis for the kernel and range of a linear operator over a polynomial field, but I'm having a little trouble with the kernel. Here's an example I made up consider ...