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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Least-squares solution to a transformation between coordinate frames

Suppose I have four coordinate frames in 3D space: A, B, X and ...
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To show that orthogonal complement of a set A is closed.

To show that orthogonal complement of a set A is closed. My try: I first show that the inner product is a continuous map. Let $X$ be an inner product space. For all $x_1,x_2,y_1,y_2 \in X$, by Cauchy-...
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recommendation for exercises/problem books

I am studying 1.Multivariable Calculus - with emphasis on limits, continuity , differentiability of functions of two variables , maxima minima 2.Linera algebra -emphasis on questions of ...
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3answers
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Why is quadratic form defined via a symmetric bilinear form?

A typical definition of quadratic form goes like this: Let $B:V\times V \to F$ be a symmetric bilinear form. A function $Q : V → F$ defined by $Q(v) = B(v, v)$ is called a quadratic form. Why ...
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$Rank(A)=$number of non-zero eigenvalues then is $Rank(A)=Rank(A^2)$?

Let $A$ be an $n$ by $n$ matrix on some field. If $Rank(A)=$number of non-zero eigenvalues of $A$ then can we say that $Rank(A^2)=Rank(A)$? I believe we can say this (thinking about idempotent ...
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Eigenvalues of Matrix Riccati Equation

Consider the Matrix Riccati Equation $$ PA + A^TP + PRP + Q = 0, $$ with positive definite matrices $Q, R$, i.e. $Q = Q^T > 0$, $R = R^T > 0$ and a negative definite $A$, i.e. $A = A^T < 0$...
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883 views

Show that the dual norm of spectral norm is Nuclear norm.

Could someone help to understand that the dual norm of spectral norm is Nuclear Norm ? We can focus on the real field. Given a matrix $X \in \mathbb{R}^{mn},$ then the spectral norm is defined by: ...
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Question about dual basis

If we have a $k$-bilinear pairing $V \times W \to k$ where $V$ and $W$ are $k$-vectorspaces of same dimension. Whenever having a basis $A$ of $V$ what does it mean to say that $B$ is a basis for $W$ ...
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Solutions to linear equations

Am I right in thinking that the following augmented matrix equation only has one solution: $\begin{bmatrix} 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 10 \end{bmatrix} $ i.e., if the ...
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If $A^n$ is normal, is $A$ normal?

My question is : Given an invertible matrix $A$ ( with complex entries ) , if $A^n$ is normal,is $A$ normal? This is related to the question : If $A$ is an invertible $n\times n$ complex matrix and ...
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Geometry software

I have some computer graphics programming experience, not so long ago, I understood that I haven't grwat experience in math and there is some problem. So, im learning math and coding at the same time. ...
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Finding the equation of the plane.

So I was going over past problems I've done and for the life of me I can't remember how to do this. Find the equation of the plane containing $$\begin{bmatrix}3 \\0 \\0 \end{bmatrix} , \begin{...
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1answer
75 views

If $A$ is an invertible $n\times n$ complex matrix and some power of $A$ is diagonal, then $A$ can be diagonalized

Prove or provide a counter-example: If $A$ is an invertible $n\times n$ complex matrix and some power of $A$ is diagonal, then $A$ can be diagonalized. If $A^n$ is a diagonal matrix, then clearly $A^...
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what does the set containing only the zero vector actually span?

I apologize if this sounds stupid but I am struggling to grasp the following concept. I understand that the span of the empty set is the zero vector. However, what does the set only containing the ...
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370 views

Dual Space Annihilator in C[0,1]

Let $V = C[0,1]$ and let U be the subspace of functions of the form $y(x) = ax+b$ for some a, b depending on the function. Give an explicit family of functionals $F\subset U^\perp$ such that for any ...
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Comparing Bases in $\mathbb{R}^{n}$

A bit of trouble with the following question: Let $\mathcal{B}$ be the basis of $\mathbb{R}^{n}$ consisting of the vectors $\vec{v_1},\vec{v_2},\cdots,\vec{v_n}$, and let $\mathcal{E}$ be some other ...
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1answer
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Why is it false that every function from $\mathbb{R}^n$ to $\mathbb{R}^m$ has a standard matrix

I ran into this problem in my Linear Algebra book while covering linear transformations: True or False: Every function from $\mathbb{R}^n$ to $\mathbb{R}^m$ has a standard matrix. I said true ...
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Solving Non Homogeneous Differential Equation, help.

My task is to solve the non homogeneous system: $x_1'=x_2+2t$ $x_2'=-x_1+2x_2-3$ with $F(t)=[2t,-3]$. It can be rewritten as: $x'=P(t)x+F(t)$ I know that to solve this I will first solve for the ...
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Dependent and independent vectors.

The indexed family $u_{1},u_{2}$ where $u_{1}=u_{2} \neq \vec{0}$ are linearly dependent ( because $u_{1}$ and $u_{2}$ are collinear) and linearly independent at the same time ! we have $\alpha ...
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1answer
854 views

Precise definition of distinct vectors

What is the precise definition of 'distinct vectors'? In particular, are the vectors (2, 1) and (4, 2) distinct, seeing as they are multiples of each other?
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513 views

Constructing the Matrix B “column by column”

I'm going through the various ways to construct a B-matrix of a linear transformation and I'm hitting a snag with one of the methods. We have $A = \begin{pmatrix} -3 & 4 \\ 4 & 3 \\ \end{...
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Linear Algebra Proof for matrices

Could someone possibly help me in proving this: Let $A$ be the augmented $m \times (n + 1)$ matrix of a system of m linear equations with $n$ unknowns. Let $B$ be the $m \times n$ matrix obtained ...
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FP3 Vectors question

$$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$$ $$\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}$$ $$\mathbf{c}=c_1\mathbf{i}+c_2\mathbf{j}+c_3\mathbf{k}$$ Use appropriate determinants ...
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Time invariace of a linear system dependent on a particular time instant

$$y[n]=x[n]+35*x[n-1]+x[0]$$ Is this system time invariant? I am under the impression that $x[0]$ can be considered a constant. Am I right?
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102 views

What does it mean to say that determinant is multilinear?

Can someone clearly explain to me what is meant by the determinant being multilinear, and what (multi-)linear functions are? I can't find a clear answer to this question.
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1answer
80 views

Jordan Normal form method

I have the matrix $A=\begin{pmatrix}0& 0& 1\\1& 0& -3\\0& 1& 3\\\end{pmatrix}$ to put into Jordan normal form. I have looked at various websites but they all seem to use ...
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1answer
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Determining properties of solution of a second order ODE

If the functions $y_1$ and $y_2$ are linearly independent solutions of $$y'' + p(t)y' + q(t)y = 0$$, show that between consecutive zeros of $y_1$ there is one and only one zero of $y_2$. Note that ...
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1answer
50 views

Any advantage when proving linear algebra statements without using bases?

I always felt that proofs in linear algebra that do not assume the existence of bases seem more elegant but is there also something mathematically more valuable about these proofs? I know that the ...
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1answer
51 views

show that the Wronskian of $x^{(1)},x^{(2)},x^{(3)}$ is identically zero or else never vanishes.

My task is to prove that: If $x^{(1)},x^{(2)},x^{(3)}$ are solutions of $X'=A(t)X$ on some interval $I$, show that the Wronskian of $x^{(1)},x^{(2)},x^{(3)}$ is identically zero or else never ...
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How can we use the symmetry of this complex matrix?

Find the Jordan normal form of $A\in \mathbb C^{4,4}$ if A is symmetric, $A^2=A$ and $\operatorname{rank} A=3$. So $A^2=A$ implies that the only eigenvalues are $0$ and $1$. From $\operatorname{rank}...
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No two 2x2 matrices in Jordan form are similar?

Let $S$ be the set of 2x2 matrices in Jordan Normal form $\begin{pmatrix}x&a\\ 0&y\end{pmatrix}$ with $a=0$ or $1$ and $x \leq y$. How do I show that no two matrices in $S$ are similar? Thank ...
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Every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix.

I would like to prove that every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix. How is this possible? I'm not sure where to begin really- I know that a nilpotent ...
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Find the eigenvalues of a projection operator

A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator. In this question is it necessary to define what the projection operator is? And won't ...
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1answer
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Finding rotate matrix which solves equation

I try to solve the following problem: given a unit vector v, find rotate matrix R such that R*v = (0,0,..0,1) (vector that it's (n-1) components are 0 and the n'th component is 1). I know that if I ...
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1answer
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Finding basis for $\operatorname{Im} T$ using $[T]_{C}^{B}$

Suppose that $T : V \to W$ is a linear map, $B=(b_1, \dots , b_n)$ - basis for $V$ and $C=(c_1, \dots , c_m)$ basis for $W$. Now, I want to find a basis for $\operatorname{Im} T$. Here's a way to do ...
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Does $B = \{x-2, x(x-2), x^2(x-2)\}$ span $\{p(x)\in P_3(\mathbb{R})|p(2) = 0\}$?

Let $P_3(\mathbb{R}) = \operatorname{Span} \{1, x, x^2, x^3\}$. $W$ is a subspace of $P_3(\mathbb{R})$, $W = \{p(x)\in P_3(\mathbb{R})|p(2) = 0\}$. Find a basis and the dimension of $W$. I chose ...
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Solving multiplicative system of equations

I am interesting in figuring out when systems with multiplication instead of addition have solutions and how to find them, for example this system: $$ \left\{\begin{matrix} a \cdot c &=& \...
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In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

$(V,\|\cdot|)$ be a real normed linear space such that $\|x\|=\|y\|$ implies $\lim\limits_{n\to\infty} \|x+ny\|-\|nx+y\|=0$, then is it true that the norm comes from an inner-product space ?
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Which determinant could we find?

$A$ and $B$ are matrices and I found the determinants of $$A + B,\, A - B,\, AB,\, A^{-1},\, B^T.$$ If we know the determinants of $A$ and $B$ but don't remember the matrices $A$ and $B$, which of ...
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Which of the following vector spaces are isomorphic? [on my last try]

So far I have tried for the first problem: A&B, B&C, A&B&C and for the second problem B&C&D, A&B&C&D, & A However, they have turned out to be wrong. I know ...
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Linear Independence of Linear Transformations

Define $V=P(\mathbb{R})$ and for $j\geq 1$ we define $T_j(f(x))=f^{(j)}(x)$ (jth derivative of $f$). We want to show that the subset $\{T_1, T_2,...T_n\}$ of the vector space $L(V)$ of linear ...
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What is the dimension of $Graph(T)$?

Let there be $T:\mathbb{F^n}\rightarrow \mathbb{F^n}$ a linear transformation, and $Graph(T):=\{\,(v,T(v))\mid v\in \mathbb{F^n}\,\}$. What is $\dim(Graph(T))$? The answer is $n$ but if $Graph(T)$ ...
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1answer
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Homogenous System

This question caught me off guard. I believe I did it the right way, but it was a bit confusing and I wanted a bit more explanation of the process and to see if what I got was accurate. This is the ...
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$T^3-4T^2+4T=0$: inference about $T$

Suppose $T$ is a linear operator on $\mathbb R^2$ such that $T^3-4T^2+4T=\theta$ where $\theta$ is the null transformation. Then, describe $T$, given that $T$ is diagonalizable. My approach: $T(...
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226 views

distributive law of addition over multiplication

In the algebra of numbers, there is a distributive law of multiplication over addition: x(y + z) = xy + xz. What would a distributive law of addition over multiplication look like? Is it a valid law ...
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165 views

Finding a Set of Basic Solutions to a Homogeneous System

$$\lceil-3,6,-6,9 \rceil$$ $$\lfloor -2, 4, -4, 6 \rfloor$$ For finding a set of basic solutions of the homogeneous system, I know it'll be $AX=0$ and begin by row reducing. The issue I run into is ...
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136 views

What is the intuition behind tensors?

Can someone please explain the intuition behind tensors? Like an example or something of the similar kind that I should keep in mind reading the theorems about it? I can't visualize it Thanks in ...
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102 views

Derivative of an L1 norm of transform of a vector.

I have to take derivative of the l-1 norm. L1 is the function R in the following expression: $$ R(\psi Fx) $$ where x is a vector, F is the inverse Fourier transform, and $\psi$ is a wavelet ...