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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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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27 views

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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9 views

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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1answer
15 views

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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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?

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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1answer
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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
16 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
28 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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1answer
34 views

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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7 views

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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4 views

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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2answers
37 views

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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2answers
38 views

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
27 views

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
15 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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1answer
25 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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1answer
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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4answers
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
50 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
38 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 ...
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1answer
15 views

How to show $S_1\subset W_1$ and $S_2\subset W_2$ are independent $\implies$ $S_1\cup S_2$ is independent based on the following assumption?

Let $W_1$ and $W_2$ be subspaces of vector space $V$ satisfying $W_1\cap W_2=\{0\}$ ,how to show $S_1\subset W_1$ and $S_2\subset W_2$ are linearly independent $\implies$ $S_1\cup S_2$ is linearly ...
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2answers
22 views

Find the matrix representing T and Find the Image of T (as a span of vectors)

Let $T(a,b) = (a+b,2a-b,3a)$. a)Find the matrix representing $T$. b)Find the image of $T$ (as a span of vectors). So I found that $T$ is a linear transformation. Now would the matrix just be $A$= ...
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1answer
14 views

Basis for Vector Space iff can be Expressed Uniquely as Linear Combo of Basis

Let $V$ be a vector space and $\beta= \{ u_1,\dots ,u_n \}$ be a subset of $V$. $\Rightarrow$ $\beta$ is a basis for $V$ iff each vector $v\in V$ can be unquiley expressed as a linear combination of ...
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2answers
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Is $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent if $S_1$ and $S_2$ are linearly dependent subsets of vector space $V$?

Let $S_1$ and $S_2$ be linearly dependent subsets of vector space $V$, are $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent? The counterexample for the first one I can think of is ...
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0answers
14 views

Computing a new inverse given information from an earlier inverse:

Suppose I have linearly independent set of vectors $v_1 ... v_k \in \mathbb{R}^{k}$. I can let $B = [v_1 ... v_k]$ and by applying Gaussian elimination on the matrix $$ [ B \ I_k] $$ I end up ...
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1answer
21 views

F is a vector space and U, V, and W are subspaces of F. Prove that $U\bigcup V\bigcup W$ is a subspace of F if and only if $U,V\subset W $.

The return is a given, but what about the other implication? We couldn't solve it in class. Sorry about the formatting, I hope it isn't an issue.
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1answer
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systems of equations with variables

I have the following problem in my homework Suppose a, b, are two constant paramaters such that the system below is consistent for any values of f and g. What can you say about the numbers a ...
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1answer
18 views

How do I get a solution set equals to a sub space?

I've four vectors that spans the $\mathbb{R}^4$ sub-space $W_1$: $\alpha_1 = \{-1,0,1,2 \}$, $\alpha_2 = \{3,4,-2,5 \}$, $\alpha_3 = \{0,4,1,11 \}$, $\alpha_4 = \{0,4,1,11 \}$ And I'm asked to ...
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0answers
21 views

why does matlab give me a negative number?

I have the following problem A steel company has four different types of scrap metal (called Typ-1 to Typ-4) with the following compositions per unit of volume They need to determine the volumes ...
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1answer
14 views

Suppose V and W are finite-dimensional. Let $v \in V$. Let $E=\{T \in \mathscr{L}(V,W)\ |\ Tv=0\}.$ Show E is subspace of $\mathscr{L}(V,W)$

Suppose V and W are finite-dimensional. Let $v \in V$. Let $$E=\{T \in \mathscr{L}(V,W)\ |\ Tv=0\}.$$ a.) Show E is subspace of $\mathscr{L}(V,W)$ b.) Suppose that $v \neq 0$, what is dim E? Here is ...
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How to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$ based on the following assumption?

Let $W=$Span$\{1, x\}$, $W_1=$Span$\{x^2, x^3\}$ and $W_2=$Span$\{1+x+x^2+x^3, 1+x+x^2-x^3\}$, how to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$? $P_3(R)=W+ W_1$ because Span$\{1, ...
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2answers
26 views

Matrix differentiation proof of quadratic product $x^TAx$

would appreciate any hints with the proof for $x^TAx$ using index notation: Suppose $x$ is an $n$ x 1 vector, $A$ is an $n$ x $n$ matrix. $A$ does not depend on $x$, and also $\alpha = x^TAx$. Let ...
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Subspace of linear transforms from V to V

Suppose V is finite-dimensional and the $\mathscr{E}$ is a subspace of $\mathscr{L}(V)$ such the $ST\in \mathscr{E}$ and $TS \in \mathscr{E}$ for all $S \in \mathscr{L}(V)$ and all $T \in ...
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Find the mapping of inverse of linear transformation

Check if $L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x)$ is a linear transformation on space of polynomials $P_2(x)$ where $p(x)=ax^2+bx+c$. If yes, find its matrix in standard basis and find the ...
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1answer
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Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify ...
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1answer
28 views

How to find $U+W$ and $U\cap W$ based on the following assumption?

Let $U=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_2=0, x_3+ x_4=0 \}$, $W=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_3=0, x_2+ x_4=0 \}$, how to find $U+W$ and $U\cap W$? I think $U\cap W=\{(x_1, x_2, ...
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How to define and make the dot product of two continuous matrix?

I was thinking recently that i always learn algebra with discret basis. But in case where the basis is continuous, how can i define a continuous matrix and when it is define how can i do the dot ...
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1answer
5 views

Solving simultaneous linear congruences for two unknowns

Find all pairs $(x,y)$ which solve $$ \left\{ \begin{align} 9x+20y&\equiv0\mod{29}\\ 16x+13y&\equiv0\mod{29} \end{align} \right. $$ So I have written this in the form ...
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1answer
28 views

Find the dimension and a basis of a subspace

Let $U$ is the set of all commuting matrices with matrix $A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}$. Prove ...
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How to show that $V=$Span$(S_2)$ if Span$(S_1)=V$ and that every vector in $S_1$ is in Span$(S_2)$?

Let $S_1$ and $S_2$ be subsets of a vector space $V$. Assume Span$(S_1)=V$ and that every vector in $S_1$ is in Span$(S_2)$, how to show that $V=$Span$(S_2)$ as well? In my opinion, to show ...
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2answers
15 views

Finding the general solution of a system of linear equations

so I've come across this question in preparation for an exam: Let $A$ be a $4\times 4$ matrix where $rank(A)=3$. The vectors $(1,2,0,-1),(0,2,1,1)$ are solutions to the system ...
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0answers
12 views

How to rotate a 3D object, using only local x-, y-, and z-rotations, so that it always faces a camera at the origin

I have been struggling with a difficult problem involving 3D rotations. I first came across this problem in a computer science context, but I've attempted to generalize it a bit before posting. (I ...
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1answer
14 views

How to show that $R^3$ is the direct sum of $W_1=$Span$(1,1,1)$ and $W_2=$Span$(\{1,0,0\}, \{1,1,0\})$?

How to show that $R^3$ is the direct sum of $W_1=$Span$(1,1,1)$ and $W_2=$Span$(\{1,0,0\}, \{1,1,0\})$? So we write it as $R^3=W_1+W_2$ because every $(x_1, x_2, x_3)\in R^3$ can be written as ...
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2answers
26 views

Linear maps of polynomials, the bases of the space and their corresponding matrix.

Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by: $$Tp(x)=(x^2p(x))',$$ for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} ...
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1answer
12 views

Transformations and Dependence

Hi, for these problems I generally get the gist of it. If you have some linearly dependent vectors $v_1, \ldots, v_m$ in $\mathbb{R}^n$ then when you transform those vectors $T(v_1), \dots, T(v_m)$ ...
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1answer
47 views

Theorems restricting the eigenvalue of a matrix

I have a square matrix $C$, whose entries I will denote by $c_{ij}$, and I would like to bound the magnitude of its eigenvalues. Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows: ...
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2answers
13 views

How to prove that $W_1\cap W_2\supset$ Span$(S_1\cap S_2)$ if $W_1=$ Span$(S_1)$ and $W_2=$ Span$(S_2)$ are subspaces of vector space?

In my opinion, let $v\in$ Span($S_1\cap S_2$) and therefore $v\in$ Span$(S_1)$ and $v\in$ Span$(S_2)$. Write $v=c_1z_1+...+ c_nz_n$ where $z_k\in S_1\cap S_2$ and $c_k\in R$. Here I am feeling I have ...