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Questions tagged [linear-algebra]

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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Question about Span in Linear Algebra

Does Sp(V)=Sp(W) mean that V=W? Intuitively I sense it does not, but I cannot find the right arguments to reason it… Could anybody help out?
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Null Space of Sum of Two Matrices is a subset or supset of null space of one

Could anyone explain how either of these can be proven? I don't see how either of these statements by themselves can be true, much less how to prove them. $$N(A+B)⊂N(A)$$ $$N(A+B)⊃N(A)$$
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Is there a special relationship between a norm on a vector space V, and the operator norm $ \mathcal{L}(V, \mathbb{R)}$?

Let $T$ be a linear operator in $\mathcal{L}(V)$. An operator norm is denoted as $||T||$, where it is the smallest $M$, such that $||T(v)||$ $\le$ $M||v||$ for any $v \in V$. A norm on the vector ...
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How to prove that if $A$ is an $n\times n$ matrix, then $rank(A^n)=rank(A^{n+1})$?

If $A$ is an $n\times n$ matrix, then $rank(A^n)=rank(A^{n+1})$. How to prove it?
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Set of Matrices with a given characteristic polynomial is compact [duplicate]

Consider the set of $3\times3$ matrices with the characteristic polynomial $x^3-3x^2+2x-1$. Is the set compact in $M_3(\mathbb{R})\cong \mathbb{R}^9$? The given characteristic polynomial has probably ...
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1answer
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Find tangets to a ellipse not centered at(0,0) that pass through point P

I must find the two tangents that pass through the point $(2,7)$ for the ellipse $2x^2+y^2+2x-3y-2=0$ all I was able to was getting $\frac{dy}{dx}=\frac{(-4x-2)}{(2y-3)}$ and therefore equalizing ...
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1answer
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Are they similar matrix

Do $\begin{bmatrix} 0&i&0\\0&0&1\\0&0&0 \end{bmatrix} $ and $\begin{bmatrix} 0&0&0\\-i&0&0\\0&1&0 \end{bmatrix} $ are similar.Is this True/false ...
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Show the spectral radius of a matrix is smaller than 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...
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1answer
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Systems of differential equations

\begin{array} { c } { \text { Solve the initial value problem } } \\ { \mathbf { x } ^ { \prime } = \left( \begin{array} { c c } { 1 } & { - 5 } \\ { 1 } & { - 3 } \end{array} \right) \mathbf {...
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Map preserves angle iff scalar multiple of isometry

How do I prove that a map preserves the angle if and only if it's the scalar multiple of an isometry. I get the "if" direction by using definition of isometry. How do I show the other direction, i.e....
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Difference between particular solution and general solution?

I have this linear system: \begin{cases} x + 2y - 4z + 5w = 1,\\ x + 3y +5z = 0,\\ y - 4z - 6w = -13. \end{cases} I am asked to find the particular and general solution of this system. I do not know ...
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1answer
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What properties of a linear map can be determined from its matrix?

I am currently taking a proofs based linear algebra course for math undergraduates. It's been almost two years since I took a more computational linear algebra course (solving matrix equations, ...
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Homogeneity does not suffice for a map between vector spaces to be linear

The following problem is taken from Sheldon Axler's book Linear Algebra Done Right, more precisely Exercise 1. from Chapter 3: Problem: Give an example of a function $f : \mathbb{R}^2 \to \mathbb{R}$ ...
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1answer
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$x = 3$ in $\Bbb Z_5$ equivalent to saying $x \equiv 3 \pmod 5$?

I basically want to confirm the title, if I wrote both of these in a test it'd be considered the same, right?
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Determining whether limit goes to inifnity exists for matrices in Jordan canonical form

Recently in class, we covered the Jordan canonical form of matrices. I'm just confused as to when the limit $\lim_n\rightarrow\infty$ exists for certain matrices expressed in such a form - if someone ...
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1answer
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Why we only need to verify additive identity, and closed under addition and scalar multiplication for subspace?

In the book Linear Algebra Done Right, it is said that to determine quickly whether a given subset of $V$ is a subspace of $V$, the three conditions, namely additive identity, closed under addition, ...
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Nonnegative determinant matrix as a sum of two squares

As one can see in this MSE question, we have $\det(A^{2} + B^{2})\geq 0$ for any real $n\times n$ matrices $A, B$. Is the converse true? i.e. if a real $n\times n$ matrix with $\det C\geq 0$, is given,...
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is $\det(A^2 + I)$ always non negative?

Obviously $\det(A^2)$ is (casework), but is the above matrix non-negative? $\det(A)\det(A) \geq 0$ as $\det(A) > 0$ or $\det(A) < 0$ yields positive when squared. However, I am not sure that ...
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3answers
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Let $\mathbb{F}=\mathbb{F}_3$ Find an irreducible polynomial of degree 2 and construct a field of 9 elements as a quotient.

Find an irreducible polynomial p of degree 2, and use it construct a field of 9 elements as a quotient. Describe the cosets in the quotient explicitly, and use them to construct the addition and ...
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Let $V=\mathbb{R}^3$ and $W=\{ (x,y,z): x+y+z=0\}$ Describe $V/W$ geometrically and contrsuct an explicit isomorphism $W^\perp \cong V/W$

For an isomorphism I let $\phi:W^\perp\to V/W$ be defined as $\phi(x)=[x]$ To show $\phi$ is injective suppose $x\neq y$ for $x,y \in W^\perp$. Since $W=\{ (x,y,z): x+y+z=0\}$, $(1,1,1)$ is the ...
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How to find the matrix represented by the polynomials $A^{12}-5A^{11}+…+3I$?

I need to find the characteristic equation of the matrix $ A = \begin{bmatrix} 2&1&1\\ 0&1&0\\ 1&1&2\\ \end{bmatrix} $ and find the matrix represented by $ A^{12}-5A^{11}+7A^{...
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4x4 Determinant

$A =\begin{vmatrix}a^2 & b^2 & c^2 & d^2\\ (a-1)^2 & (b-1)^2 & (c-1)^2 & (d-1)^2 \\ (a-2)^2 & (b-2)^2 & (c-2)^2 & (d-2)^2\\ (a-3)^2 & (b-3)^2 & (c-3)^2 &...
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1answer
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formula for getting the normlized X and Y values of a given degrees from a linear function

I have a number which we'll call α in degrees that represent the angle of a linear function with the X axis. for example when α is 0 the linear function is on the X axis , when α is 360 the linear ...
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2answers
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Is this $3\times3$ matrix diagonalizable?

After browsing through similar posts, I was wondering if I am understanding the meaning of "$n$ distinct eigenvalues" for the following theorem. If the $n\times n$ matrix $A$ has $n$ distinct ...
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3answers
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Finding the sec. Eigenvector, when knowing the first Eigenvector and Eigenvalue

So here is my problem, Let A be a symmetric Matrix (2x2) with EV=(-2, 3) and the Eigenvalue being -5. Find the 2. Eigenvector to the second Eigenvalue. The only info i can think of is that the EV ...
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Given that a symmetric Matrix with only two eigen values and eigen space E_{2}, Is it possible to find the entries in the Matrix

I tried to solve this using linear equations, but then I strongly sense that there is something wrong with the question.
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A Bound on the Dimensions of Certain Types of Subspaces

Let $V$ be a $4$-dimensional vector space over the complex numbers, and let $S$ be a subspace of the endomorphisms of $V$ such that the elements of $S$ commute. If there exists an element in $S$ ...
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What a pair of bases must meet in order for the matrix of f in that pair to be of the type..?

$$f:S\rightarrow V$$ $$A=\left(\begin{array}{cc} 0 & 2\\ 0 & 1 \end{array}\right)$$ Where S is the set of symmetrical 2x2 matrices and V the set of 2x2 matrices. Also $f(M)=2AM-trace(M)·A$ ...
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Finding Minimal polynomial from Characteristic polynomial

If $T$ is a linear operator with characteristic polynomial $(x^2-1)^6$ such that $\mathrm{rank}(T-I) = 9$, $\mathrm{rank}(T-I)^2 = 7$, $\mathrm{rank}(T +I) = 10$ and $\mathrm{rank}(T +I)^2 = 9$, find ...
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2answers
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Skew-symmetric non-diagonalizable matrix

Do you have an example of a real skew-symmetric matrix (seen as an operator over $\mathbb{C}^n$) having at least one (purely imaginary) eigenvalue with algebraic multiplicity strictly greater than the ...
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Inverse of the adjugate operation

In projective geometry, the map between a primal and dual quadric is the adjugate: $adj(Q) = Q^*$. The map from dual to primal is then the inverse adjugate, $Q = adj^{-1}(Q^*)$, as in this paper. ...
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Can any $n$ number of linearly independent vectors build the basis of a vector space on $\Bbb R^n$?

I'm trying to understand how many bases there can be in a vector space. According to my understanding so far, for example in the vector space $\Bbb R^2$ every pair of linearly independent vectors can ...
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1answer
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What is the relationship between the norm of a vector space and its linear functional operator norm?

I'm trying to answer the question below. I honestly cannot see any connection other than that both norm are measuring "size" in some way, which is just definitionally true. Let $V$ be a be a normed ...
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True or False ( Invertibility of $A$)

Let $A$ be $10 \times 10$ matrix such that $A^2 + A +I = 0$ then the matrix is invertible. $A^2 + A = –I$ $A(–A–I) = I$ So, – $A$ – $I$ is inverse of A which is clearly $10 \times 10$ matrix, since $...
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If $\lambda$ is eigenvalue of non-singular matrix A, show that | A | / $\lambda$ is eigenvalue of adj A. Can anyone help me out? . [on hold]

(https://i.stack.imgur.com/QXW22.jpg) Please help me with this question. I'm having an exam. I wanted to solve but I couldn't.
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prove that 2 vectors is linearly independent in the vector space of vectors of length 2 with entries of real-valued functions

I have the following vectors $V_1=(e^t,te^t), V_2=(1,t)$ now I want to prove that this two vectors are linearly independent in the vector space of vectors of length 2 with entries of real-valued ...
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1answer
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Show that $L_A$ acts on by orthogonal transformation and in particular rotation.

Let $A$ be a $3\times 3$ orthogonal matrix with determinant $=1$. Let $v$ be an eigen vector corresponding to $1$ of $A$.Let $W=\text{span}\{v\}$. Show that $L_A$ preserves $W^\perp$ and it acts ...
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1answer
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Show that $(\mathbb{R},\mathbb{R},\odot,\oplus)$ is a vector space if $\odot$ and $\oplus$ are defined by:

Show that $(\mathbb{R},\mathbb{R},\odot,\oplus)$ is a vector space if $\odot$ and $\oplus$ are defined by $\alpha \odot x = \alpha^7 (x-3) + 3$ $x \oplus y = (\sqrt[7]{x-3} + \sqrt[7]{y-3})^7+3$ ...
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1answer
22 views

Product of a diagonal matrix and an orthogonal matrix

I am in a situation where I need to prove a property, and I need to know this: Is the product of a diagonal matrix and an orthogonal matrix commutative?
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Determine the values for a for which the matrix is diagonizable.

"Determine the values for a for which the matrix A is diagonizable. " $$ A = \begin{bmatrix}1&1\\a&1\end{bmatrix}$$ My first attempt solving this problem was to find the characteristic ...
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1answer
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Mappings to spaces with different numbers of dimensions

The first diagram first question second question I seem to have a good understanding of linear transformation and linear algebra yet I fail to grasp this completely. Question 1) Please explain ...
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Transform a common matrix into a form where sum of each rows and columns 1

Is it possible to create a transformation $T$ for a matrix $A$, so a new matrix $C := T(A)$ will have sum for all rows and columns equal to $1$? Can it help if the original matrix $A$ is symmetric?
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1answer
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Distance from eigenspace of matrix

In linear algebra, is there a separate name / concept for the notion of distance between linear vector subspaces? I'm asking this because I'm considering a problem in numerical linear algebra where ...
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1answer
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Doubt about the rank-nullity theorem involving a (block) matrix

Let $P\in\mathbb{R}^{n\times n}$ and $Q\in\mathbb{R}^{n\times m}$ and define $$ S:=[Q, PQ, P^2Q, \ldots, P^{n-1}Q]. $$ Then $S\in\mathbb{R}^{n\times(mn)}$. Now, the book says that, if $\operatorname{...
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Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
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Euler angle to Multivector

What’s the formula to go from euler angles over to Multivector, and are they the same as rotors? I have some familiarity with quaternions but I don’t like that they have to divide the angles by 2 ...
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1answer
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Bounds for the rank of the sum of two linear maps

The following is Exercise 3.15 from the German textbook Lineare Algebra by Hans-Joachim Kowalsky and Gerhard O. Michler: Let $\varphi$ and $\psi$ two linear maps from a finite-dimensional vector ...
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Proof of Logarithm map formulae from $SO(3)$ to $\mathfrak {so}(3)$

According to exponential map, there also exist a logarithm map $$\log:SO(3) \to \mathfrak {so}(3).$$ Suppose a vector $t \in\mathfrak {so}(3)$ and $t=\|t\|w$, according to exponential map $$R = \cos\|...
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1answer
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Check whether a subset of a vector space is a subspace

$V = \mathbb{K}^n$, where $\mathbb{K}$ -- field. $V_1 = \{(x_1,\cdots,x_n)\in V\mid \sum_{i=1}^{n} a_ix_i = 1; a_1,\cdots a_n \in \mathbb{K}\}$. So, I should check that $V_1$ -- subspace. At this ...
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46 views

The relation between eigenvalues and inner product of eigenvectors of a real orthogonal matrix where $[R^T]=[R^{-1}]$.

what is the relation between eigenvalues and inner product of eigenvectors of a real orthogonal matrix where $[R^T]=[R^{-1}]$. (Transpose is equal to its Inverse)