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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Additivity Implies Homogeneity of Rational Scalars

I did my best to search for this question on the site- but I did not find it. Here it is: If a function $f:\mathbb{R}^2\to\mathbb{R}$ satisfies $f(u+v)=f(u)+f(v)$ for all $u,v\in\mathbb{R}^2$, ...
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If a subspace of a finite-dimensional vector space. Then the subspace is finite dimensional?

I have difficulty in understanding the proof of this statement: Let W be a subspace of a finite-dimensional vector space V. Then W is finite dimensional. The proof goes like this. (Linear algebra, ...
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24 views

Area of a parallelogram with three points in $\mathbb{R}^{n}$: $(a,b, 0); (a, 0, b); (0, a, b)$

I have been requested to calculate the area of the parallelogram with three adjacent vertices: $(a,b, 0); (a, 0, b); (0, a, b). First, I have made this diagram: Then I proceed to calculate the two ...
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Condition for right handed invertibility

Suppose that $A$ is an m by n matrix and is right invertible, such that there exists and an n by m matrix $B$ such that $AB = I_m.$ Prove that $m\leq n.$ I'm not really sure how go about this ...
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Find the image of the transformation and write as a span of vectors.

Let $T(a,b)=(a+b,2a-b,3a)$ Find the image of $T$ (as a span of vectors). So I created the augmented matrix and got this: $A$= $\begin{bmatrix}1 & 1 & b_1\\2 & -1 & b_2\\ 3 &0 ...
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How can I show that for matrix $A$ , $A^t A $ is not equal to $ A A^t $ in general?

How can I show that for matrix $A$ , $A^t A \neq A A^t $ $A^t$ means the transpose of $A$. That is the entire question and I have no idea how to begin... please help!
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What are the $GL_n(F)$-orbits of a group action on the set of idempotent matrices?

Let $S= \{A \in M_{n \times n}(F):A^2=A\}$ (set of idempotent matrices). The general linear group $G=GL_n(F)$ acts on $S$ by $A.g=gAg^{-1}$ (conjugation). I'm having trouble visualizing the ...
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Find $\phi, \psi: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, each nilpotent of order 2, whose composition is idempotent

$\phi: V \rightarrow V$ is nilpotent of order 2 if $\phi \phi$ is the zero endomorphism. Now composition of two such endomorphisms need not be nilpotent of order 2. Find $\phi, \psi: \mathbb{R}^2 ...
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Range of sum of vector space

Let $S,T$ be elements of $L(V,W).$ Show that the range$(S +T)$ is a subspace of range$(S)$ + range$(T)$. I tried applying the definition of range, but I wasn't sure how to proceed after that.
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$e^{(A+B)} = e^Ae^Be^{[A,B]}$ for non commuting A and B?

For non commuting A and B, and the derivative of $[A,B] = 0$. Is it true that/how to prove that $e^{(A+B)} = e^Ae^Be^{[A,B]}$ If not, what is the expression according to Wikipedia's article on the ...
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Proving a norm is lipschitz

Let $M\in\mathbb{R}^{n\times n}$. Define the function $f\colon\mathbb{R}^n\to\mathbb{R}$ by $f(x)=\Vert Mx\Vert$. Show that $f$ is Lipschitz. Let $x,y\in\mathbb{R}^n$, then we want to find a ...
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Derivation of variance of a linearly transformed vector

I am trying to derive the variance of a linearly transformed vector. A result was given here. $$ \mathbf{y} = X \, \mathbf{b} $$ $$ \mathbf{b} \sim \mathcal{N}( \mathbf{0}, \sigma^2 I) $$ If we say ...
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Image and Kernel of a Projection of One Line onto Another

The question is: Let T be the projection along a line L1 onto a line L2. Describe the the image and the kernel of T geometrically. I understand that the image should be the Projection of L1 onto L2. ...
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31 views

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$ with $X_i$'s $ \in \mathbb{R}$ Just from computing $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$ I am guessing the general formula is: $(x_1 + \cdots + ...
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Determinant of an elementary matrix

I read a very slick proof of determinant properties, in this case of the fact $\det A = \det A^T$, which says in one place It suffices to notice that for any elementary matrix $M$ we have $\det M ...
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Understanding a proof of RREF uniqueness

Base Case $(n = 1)$: Suppose $A$ has only one column. If $A$ is the all zero matrix, it is row equivalent only to itself and is in reduced row echelon form. Every nonzero matrix with one column has ...
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What is the relation between the cokernel with the kernel of the dual map of a linear transformation?

I am studying linear algebra and I am in front on questions like: What is the relation between the kernel of a linear map and the cokernel of the dual map? What is the relation between theese objects ...
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Define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine ker$\phi$ and im$\phi$

Let {$e_1,e_2,e_3$} be the standard basis for $\mathbb{R}^3$ and define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine the subspaces ...
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A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
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Show that $Im T$ and $U/Nuc T$ are isomorphic for a linear transformation $T: U \longrightarrow V$

Show that $Im$ $T$ and $U/Nuc$ $T$ are isomorphic for a linear transformation $T: U \longrightarrow V$ Hi guys, I know how to show this for vectorial spaces with finite dimension, but I don't have ...
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How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans?

Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$ Apparently $\dim W_1=\dim W_2=2$. For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as ...
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1answer
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How to construct an isomorphism between $ \ker g^{\ast}$ and $~coker~ g$?

Let $g: L \to M$ a linear transforming. $M, L$ finite dimensional. $g^{\ast} : M^{\ast} \to L^{\ast}$ How do I construct an isomorphism between $ \ker g^{\ast}$ and $coker~ g$? I really don't know ...
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32 views

Show that $P_n$ is an $(n+1)$-dimensional subspace of the vector space of all real polynomials [duplicate]

Show that $P_n$ = {polynomials with real coefficients of degree $\leq$ n} is an ($n+1$)-dimensional subspace of the infinite-dimensional vector space of all real polynomials I know that $P_n$ is ...
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1answer
22 views

choosing a square matrix to have a product with one 1 und other 0's

Let $A$ be a $m\times n$ real matrix with maximal rank. Let $i\in\{1,\dots,m\}$, $j\in\{1,\dots,n\}$. I'm curious if it is possible (for any choice of $i,j$) to find a square matrix $B$ such that ...
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25 views

Dimension of the image of a matrix

So the question asks: Verify if the image of the linear map $T : \mathbb{R}^6 \to \mathbb{R}^3$ given by left multiplication by A= $$\begin{bmatrix}6 & 0 &2 & 2& 3& 4\\0 & -1 ...
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Understanding a problem

Note that these from linear algebra notes. İt was defined fields, showed $\mathbb{Q}$ is a field. Then, below-mentioned qustion was proved. Yet, I didn't ask what happened. Can you explain? What ...
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Number of eigenvalues for this operator

Say I have a F - vector space V and a subspace U given by U={va : a is in F}. Now suppose I have an operator defined by $Tv=av$. Clearly, U is invariant under T, since for any element of U, say bv, I ...
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Unit vectors with imaginary numbers

I'm trying to determine if the matrix: \begin{bmatrix} 0 & i \\ 1 & 0 \end{bmatrix} is a unitary matrix. Therefore, the first step I'm taking is to figure out if both $\langle 0, 1\rangle$ ...
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Probing that $\sin^2{\phi}+\cos^2{\phi}=1$ for cross and dot product

I have this problem statement: Use the cross product to find the sine angle $\phi$ between the vectors $\vec{u}=2i+j-k$ and $\vec{v}=-3i-2j+4k$. Then use the dot product to find the cosine angle ...
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Find matrix of composition of linear transformations

Let $f : \mathbb R^4 \to \mathbb R^{2x2}$ and $g : \mathbb R^{2x2} \to P^2$ ($P^2$ - all polynomials of degree 2 max) be linear transformations, given by: ...
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Find the value of $||T||$ if T is defined as:

This question was asked in GATE 2016: Please help me to figure out the right answer. Let $T$ ∶ $ℓ_2$ → $ℓ_2$ be defined by $T((x_1,x_2,...,x_n...))$=$(X_2-X_1, X_3-X_2,...,X_{n+1}-x_n,...)$ Then (A) ...
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0answers
23 views

Linear transforms and their corresponding invertible matrix.

Let $(1,x,x^2,x^3)$ be a basis for $\mathscr{P_3}(\mathbb{R})$ and let $(1,x,x^2,x^3,x^4)$ be a basis for $\mathscr{P_4}(\mathbb{R})$. Suppose $R \in ...
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Is it always possible to find the Reduced Row Echelon form of a matrix, given the basis of its null space? [on hold]

I tried starting with multiple bases of the null space and each time I was able to write the RREF form of the matrix. However, I have not been able to prove that this is true for all possible bases.
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cauchy- schwarz inequality b/2a input value

I was watching this video but at 8:05 I don't get why to solve for the function $p(t) = at^2 + bt + c \geq 0$, Sal decides to input $t= \frac{b}{2a}$. Someone made this explanation: $\frac{b}{2a}$ is ...
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1answer
19 views

Normal real matrix

Q: Is there a normal, real matrix, which it's characteristic polynomial is $t(t-1)(t^2+1)$ ? I think that there isn't such matrix, and I prove it by: if a matrix is real than it's diagonalizable, ...
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1answer
32 views

When is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible?

The question is quite simple: for a $N \times p$ matrix $\mathbf{X}$ with real entries, when is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible (where $\mathbf{I}$ is the $p \times p$ identity ...
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Testing the diagonalizability of matrix $B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$

How to show that the matrix $$B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$$ is diagonalizable when $a\neq0$, when ...
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Matrix transformation into block off-diagonal form

Consider the 4-by-4 matrix $\boldsymbol M = \boldsymbol M_0 + \boldsymbol M_1$, where $\boldsymbol M_0 = \alpha \left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 ...
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Let $(L_1,L_2,L_3)$ be an ordered triple of pairwise distinct plane in $K^3$. There is two possibles types of relative arrangement of such triples.

Let $(L_1,L_2,L_3)$ be an ordered triple of pairwise distinct plane in $K^3$. Prove that there is two possibles types of relative arrangement of such triples characterized by the fact that $\dim ...
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Prove that the two vectors have to be linearly independent

Say you have three vectors $u,v$, and $w$ in $\mathbb{R}^3 $ that are linearly independent. Prove that the two vectors $u+w$ and $v+w$ have to be linearly independent. (start by assuming ...
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What is a basis and dimension of $span\{I,M,M^2,…\}$ where $I$ is the identity matrix and $M$ is invertible squared matrix?

Putting all vectors (matrices) in one gives $$ \begin{bmatrix} 1 & 0 & 0 & m_1 & \cdots\\ 0 & 1 & 0 & m_2 & \cdots\\ 0 & 0 & 1 ...
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1answer
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Showing that non-diagonalizable matrix is similar to upper triangle matrix

I have the following task: Let $A\in \mathcal{M}_3(K)$ be a non-diagonalizable matrix where $K$ is a field and the characteristic polynomial of $A$ is ...
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1answer
16 views

Quadratic form inequality implies matrix inequality?

Suppose we have the following quadratic form: $$ x^T(t)(A^TP+PA)x(t)\le-x^T(t)Qx(t)\quad\forall t $$ where $P$ and $Q$ are symmetric positive definite matrices and $\dot x(t)=Ax(t)$. Why does the ...
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28 views

Centralizer of $A$ is equal to $\langle A \rangle$

Let$$A=\begin{pmatrix} 0 & a \\ 1 & b \end{pmatrix}.$$ How to prove or disprove that the centralizer of $A$ is equal to $\langle A \rangle$ (matrices generated by A)? For a matrix to be in ...
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60 views

Does $AB=(AB)^{\ast}$ and $A=A^{\ast}$ implies $B=B^{\ast}$?

Suppose that we have $AB=(AB)^{\ast}$ and $A=A^{\ast}$, does this implies that $B=B^{\ast}$? ($A^{\ast}$ is the Hermitian adjoint of $A$.) I have a feeling that they might not be equal in general. ...
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3answers
32 views

Scaling a matrix to make its eigenvalues fall within a certain interval

Suppose I have a diagonalizable matrix $M$ which has all its eigenvalues between $a$ and $b$. Is it possible to scale $M$ to $M_S$ such that all the eigenvalues of $M_s$ lie in the interval $[-1,1]$? ...
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1answer
33 views

How do I prove that for $\|T(x)\|=\|x\|$ for all $x$ in a vector space iff $\left<T(x),T(y)\right>=\left<x,y\right>$?

Cleaner version: $$\left \| T(x) \right \|=\left \| x \right \|,\forall x\in V\Longleftrightarrow \left \langle T(x),T(y) \right \rangle=\left \langle x,y \right \rangle$$ I know that $\left \langle ...
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61 views

Does a definite integral define a linear functional? [on hold]

Would $\displaystyle\int_0^1 t^2x(t)\,dt$ be a linear functional? For each $x$ in $P$ the function $y$ is defined by $\displaystyle\int_0^1 t^2x(t)\,dt$. I have to show that $y(ax+bz) = ay(x)+by(z)$. ...
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1answer
51 views

group action same thing as homomorphism

A linear group action of a group $G$ on a vector space $V$ is the same thing as a homomorphism from G to the general linear group $GL(V)$. attempt: Suppose a linear group action of a group $G$ on a ...
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4answers
39 views

Eigenvector of a matrix of ones associated with $\lambda =0$

An $n\times n$ matrix consistent of all ones, will have two eigenvalues: $0$ and $n$. The eigenvector associated with $n$ will be $(1,1,...,1)$, but are there then infinite solutions for the ...