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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Numerical Analysis: Matrix Norm comparison

It's my first time posting on this site so I'll be quick :). I need help understanding and solving this problem. ...
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Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
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Finding a linear transformation with a given null space

The problem statement is, Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$. ...
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Relating regression to projection?

I recently learned that one can think of regression as a projection of a vector in a high dimension space onto the other vector. I tried implementing this and got it to work: ...
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Why do I have to show this subspace is an invariant subspace?

Consider a vector space $V \cong \mathbb{R}^n$ with an operator $I \in O(n)$ satisfying the property $I^2 = -Id_{V}$. See Linear Complex Structure for context. I want to show that $V$ has real ...
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Show that an inverse of a bijective linear map is a linear map.

So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well? enter image description here
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Dimension of the subspace of a vector space spanned by the following vectors.

I know that in order to find a subsequence that is a basis of a subspace is to check whether the given vectors are linearly independent and whether they span the subspace. However how can I find the ...
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15 views

A question concerning isometries and determinants

Let $e_1 = (1, 0)$ and $e_2 = (0, 1)$. Assume that $f : \Bbb R^2 → \Bbb R^2$ is an isometry of the plane fixing $(0, 0)$. Let $f(e_1) = (a, b)$ and $f(e_2) = (c, d)$, and let $A = \begin{vmatrix} ...
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What's the period of this matrix?

Consider the matrix $$ A = \begin{pmatrix} 0.1 & 0.3 & 0.4 & 0.2 \\ 0.2 & 0.4 & 0.0 & 0.4 \\ 0.0 & 0.3 & 0.5 & 0.2 \\ 0.5 & 0.3 & 0.2 & 0.0 ...
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Finding the basis of a subspace

I understand that the basis of a subspace defined by this equation requires you to find a combination of $x_1,x_2,x_3$ that satisfy this equation [so $(-1,0,2)$ for example]. But how do you know how ...
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Proving that if $A$ is diagonalisable then $\chi_A(A) = 0$

This could be a very simple question to answer, but I'm unsure how to prove this. If you have a diagonalisable matrix $A$, prove that $\chi_A(A)$ is the zero matrix. (where $\chi_A(x)$ is the ...
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Question about how the determinant of a square matrix can help determine whether a set of vectors is a basis.

I have a linear algebra midterm tomorrow. While it's highly unlikely a question of this type shows up, I really wanted to understand this because I am curious since I've spent so long without coming ...
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1answer
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Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...
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$a+b$ for $ax+3y=5$ and $2x+by=3$

If $ax+3y=5$ and $2x+by=3$ represent the same straight line, then what does a+b equal? I've tried this, $ax+3y=5$ and $2x+by=3$ Multiply to equal 15 so they equal each other ...
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1answer
19 views

$C_o= \{(x_n):x_n \in R, x_n \rightarrow 0 \}$ and $ M=\{(x_n):x_n \in C_0,~~ x_1+ x_2+…+x_{10}=0\}$ then dimension of $(\frac{C_0}{M})$ is equal to

Let $C_o= \{(x_n):x_n \in R, x_n \rightarrow 0 \}$ and $ M=\{(x_n):x_n \in C_0,~~ x_1+ x_2+...+x_{10}=0\}$ then dimension of $(\frac{C_0}{M})$ is equal to
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Conditioning in regards to matrix vector product

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
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1answer
13 views

Independance, Inverse and Additive Identity for vector with defined vector addition and scalar multiplication

I am struggling to find my bearings on this question. I am confident that I can do parts a and d. I have no clue how to approach b. I was also wondering if the redefined vector addition and scalar ...
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23 views

About a particular definition of “tensor”

I came across this quiet new to me way of defining "tensors", That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n ...
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1answer
22 views

How to tell that $W$ is a subspace of $ \mathbb R^3$?

To do this problem, I wrote this matrix in RREF form and found that $V_3$ is $-1V_1 + 2V_2$. This demonstrates that these planes are a basis for $ \mathbb R^2$. However, I am not sure to extend that ...
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normal operator equation

let $S: V \to V$ linear transformation in a inner product space of $\mathbb{C}$. Prove that $S$ is normal iff $$\|S(x)\| = \|S^*(x)\|$$ That's what I have done so far: if $S$ is normal than $$SS^* ...
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Prove that elementary matrices perform row operations

How to prove that elementary matrices actually perform their intended row operations: multiplying by a constant, adding a multiple of one row to another, and switching two rows? I've seen examples ...
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$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
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What is the meaning of the notation [A|B] in Linear Algebra.

I am going through Linear Algebra right now, we are using the book Elementary Linear Algebra by Andrilli. In one of the theorems he uses this notation without really introducing it. Here is the ...
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19 views

If $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$

Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then: if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary. if $A$ is normal and $f_A(x)=f_B(x)$ ...
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13 views

Write $F$ as a linear combination of elements of $\mathcal B^*$

If $V=\mathbb R[x]_k=\{\sum\limits_{i=1}^ka_ix^i:a_i\in\mathbb R, \forall i\}$ is a vector space of dimension $k+1$ over $K=\mathbb R$ and $\mathcal B=\{1,x,\dots,x^k\}$ is a basis of $V$. The dual ...
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How to change a basis of some orthogonal operator to obtain canonical form of operator [on hold]

We have a linear operator $f:X\rightarrow X$ on 3-dimensional real inner product space $X$ which has in O.N. basis $e_1, e_2, e_3$ a matrix $$ \left [ \begin{array}{rrr} \cos t & \sin t & 0\\ ...
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Linear Algebra - Transformations, image, kernel [on hold]

Question Define T : R3 → R3 by Tx = (x · (1, 0, −1))(1, 0, −1) + (x · (1, 1, 1))(1, 1, 1) (a) Compute the action of T on the unit vectors i, j, k. (b) Write down the standard ...
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For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R\{1}. [on hold]

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R \ {1}. (a) Show that a ∗ b ∈ G for all a, b ∈ G. (b) Show that G together with the binary operation G × G → G, (a, b) |→ a ...
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The formula $\DeclareMathOperator{tr}{tr}\mathrm{adj}(A)=\tfrac{1}{2}[(\tr A)^2-\tr(A^2)]I_3-[\tr A]A+A^2$ for the adjoint of a $3\times 3$ matrix

Let $A$ be a square matrix of order $3$. Prove that $$\operatorname{adj}(A)=\frac{1}{2}[(\operatorname{tr} A)^2-\operatorname{tr}(A^2)]I_3-[\operatorname{tr} A]A+A^2$$ where $\operatorname{tr}A$ is ...
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Describing all the linear transformations satisfying the constraints

How to find the linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satistfying $4x_1-3x_2+x_3=0$ is a) Null space of $T$ b) Range of $T$ I'm not able to ...
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Linear Algebra - Transition matrices

Question I have some methodological questions with this exercise: 1. You are given that the transition matric $P_{\mathcal C,\mathcal B}$ from a basis $\mathcal B=\{b_1,\ b_2,\ b_3\}$ to a basis ...
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Examine if the set is linearly independent

How do I prove or disprove if $\{1, \cos x, \cos 2x,..., \cos nx\}$ is linearly independent? I tried solving the problem using the definition of linear independence, $\sum_{k=0}^n a_k\cos kx = 0$ ...
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Awkwardly formed linear spaces exercise

I came across such an exercise: Let $V$ be a linear space over $K$ such that $\dim V = n$. Show that for any $\alpha_1, \alpha_2, \dots, \alpha_m$ with $ m > n + 1$ there exist $a_1, \dots, ...
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How to find unknowns $w_1,w_2,w_3$ that satisfy $t=w_1f_1 + w_2f_2 + w_3f_3$?

For any $i \in \{1,2,3\}$, let: $w_i \in [0,1]$ is an unknown number such that $\sum_{i \in \{1,2,3\}} w_i = 1$. $t$ is a known number in $[0,1]$. Suppose that $t = 0.8$. $f_i$ is also a known ...
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how to determine a matrix has a single eigenvalue

Find the jordan form of the matrix $$A = \begin{pmatrix} 1 & 1 & 2 & 2\\ 1 & -2 & -1 & -1\\ -2 & 1 & -1 & -1\\ 1 & 1 & 2 & 2 \\ \end{pmatrix}$$ ...
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1answer
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Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
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Linearity and invertibility of $A^{-1}$

If $A\in L(X)$ then prove that $A^{-1}$ is linear and invertible. Proof: Since $A$ is invertible then $A$ is injective and surjective. We know that $A^{-1}$ defines by $A^{-1}(Ax)=x$. Remark: Also ...
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Symmetric Matrix with Positive Eigenvalues

Not all matrix with positive eigenvalues is positive definite, i.e. $\mathbf{x}^\mathsf{T}A\mathbf{x}>0$ for all non zero vector $\mathbf{x}$. For example consider matrix $$A = \begin{bmatrix} 1 ...
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Show that $B^TAB$ is symmetric. [on hold]

$A$ is invertible, but it does not say that $A$ is symmetric. By $B^T$ I mean that $B$ is transposed.
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1answer
25 views

Is the the statement is true or false? [on hold]

Suppose $A$ is a $m \times n$ matrix and $V$ is a $m \times 1$ matrix with both $A$ and $V$ having rational entries and suppose the system $AX=V$ has a solution in $\mathbb{R}^n$. Then the equation ...
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1answer
17 views

If $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$

In one book on differential equations and dynamical systems I read that if (1) $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then (2) $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$. ...
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1answer
31 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
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1answer
23 views

$U,W$ are subspaces. show $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap W\}=\{U,W\}$

This is a question from a review package that is causing me some trouble. Let $U,W$ be subspaces of a finite dimensional vector space. Show if $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap ...
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1answer
21 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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Finding a Matrix B by knowing its Kernel is the Image of Matrix A

I understand how to find the image($A$). The basis of Im($A$) would be the first two columns of the matrix $A$ (given the two leading 1's in ref are in the first and second columns). So the ...
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29 views

Understanding a basic matrix theorem

There's a theorem in Linear Algebra which says that if ${\bf A}$ is an $m \times n$ matrix and $m < n$, then the homogeneous system of linear equations ${\bf A}{\bf X}=0$ has a non trivial ...
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3answers
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Showing that span$\{x,Ix\}$ is an invarient subspace of $V:=\mathbb{R}^n$

Let $V := R^n$ be a vector space and let $I \in O(n)$ be an operator satisfying $I^2 = -Id$. I want to show that the $span\{x,Ix\}$ is an invarient subspace of $I$. Let $W = span\{x,Ix\}$. I need to ...
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1answer
21 views

Proving a basis spans $R^3$

Doing some reviewing and I'm not 100% sure if my thought-process is correct. I have the following two vectors and need to prove they're a basis for $R^3$: $$B= \begin{bmatrix} 1 \\ ...
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1answer
18 views

Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space. Prove $\exists \{f_1,\ldots,f_n\}: (e_i,f_j)=\delta_{ij}$

Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space $V$. Prove there exists vectors $\{f_1,\ldots,f_n\}$ such that $(e_i,f_j)=\delta_{ij}$. I tried using ...
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42 views

Is $\{(x,y,z) \in \mathbb{R}^3 : x^2+3y^2+12z^2=0\}$ a vector space?

Is $\{(x,y,z) \in \mathbb{R}^3 :x^2+3y^2+12z^2 = 0\}$ a vector space? My inclination is that the only real solution to $x^2+3y^2+12z^2=0$ is $(0,0,0)$, which is the trivial subspace of ...