2
votes
0answers
24 views

Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Strang P59]

From P59 of Intro to Lin Alg, 4th Ed by Strang This question is not a duplicate of the original, in which user Shuchang proved the question. Presently I'm asking about further intuition or a ...
2
votes
2answers
53 views

Why there is this relation between $k$-vectors and $k$-forms?

I've been trying to understand the geometrical meaning of $k$-vectors and $k$-forms on some vector space $V$ of finite dimension $n$ over a field $\Bbb K$. Indeed, as I understood, a $k$-form $\omega ...
0
votes
0answers
34 views

How small can we make a modulus and still perform linear algebra on these pairs?

We can work with numbers of the form $(a^n + a^m)$, where $a$, $n$, and $m$ are all naturals, and $-v \le m \le v$ and $-v \le n \le v$. There is one more possibility: $a^n$ could be replaced by $0$, ...
1
vote
1answer
35 views

Motivation for Conjugate transpose of a matrix

I'am currently going through a self study of Linear algebra . I'am finding it difficult to grasp the intuition behind the concept of Conjugate transpose of a matrix .Why take the complex conjugate of ...
1
vote
0answers
294 views

How Would Arnold Explain the Jordan Normal Form to a 6 Year Old?

How would Vladimir Arnold explain the Jordan normal form, to a six year old, in full detail starting from nothing in a way that somehow explains everything in a deeper way, probably including topology ...
8
votes
1answer
179 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
1
vote
2answers
56 views

Visualise all vectors perpendicular to one vector and two vectors in R^3 [Strang P19 1.2.6]

I'm only asking about visual/geometric solutions herein. (b) The vectors perpendicular to any vector in $\mathbb{R^3}$ lie on what?. (c) The vectors perpendicular to any two vectors in $\mathbb{R^3}$ ...
2
votes
0answers
28 views

When do Entries Remain after and despite Matrix Multiplication? [Strang P92 2.5.41]

Suppose $E_1, E_2, E_3$ are 4 by 4 identity matrices, except $E_1$ has $a, b, c$ in column 1 and $E_2$ has $d, e$ in column $2$ and $E_3$ has $f$ in column 3 (below the $1$ s). Multiply $L = ...
3
votes
1answer
45 views

All Subspaces of $\mathbb{R^4}$ and $\mathbb{C^n}$ [Strang P129 3.1.14]

The subspaces of $\mathbb{R^n}$ are $\mathbb{R^4}$ itself, three-dimensional planes $\mathbf{n \cdot v = 0}$, two-dimensional subspaces $\mathbf{n_1 \cdot v = 0}$ and $\mathbf{n_2 \cdot v = ...
1
vote
0answers
35 views

Necessary and Sufficient Conditions about the Fundamental Four Subspaces [Strang P143 3.2.34]

I'm trying to ascertain the necessary and sufficient conditions on $A$ and $B$, given that $\mathbf{x \in \mathbb{C^n}}$, for : $1. \, null(A) = null(B), \; 2. \, colspace(A) = colspace(B), \; 3. \, ...
2
votes
0answers
40 views

Relationships between Reduced Row Echelon Form and the Fundamental Four Subspaces [inspired by Strang P143 3.2.34]

I'm trying to apprehend all the links between two matrices' RREFs and their $4$ fundamental subspaces. Does $RREF(A) = RREF(B) $ $1.1.$ $\implies null(A) = null(B)$? True because $null(A) = ...
3
votes
1answer
231 views

Intuition - If $Ax = b$ has infinitely many solutions, why can't $Ax = c$ have only one solution? [Strang P165 3.4. 22]

If $\mathbf{Ax = b}$ has infinitely many solutions, why is it impossible for $\mathbf{Ax = c}$ (where $\mathbf{c}$ is a new right side) to have only one solution? Proof : Take two solutions of ...
1
vote
1answer
23 views

Matrix Multiplying Column Vectors of Another Matrix = Matrix?

This question is predicated upon this answer and the comments thereunder: $$\color{green}{ \begin{bmatrix}1&2\\3&4\end{bmatrix} }\cdot \begin{bmatrix}1&2\\3&4\end{bmatrix} ...
2
votes
2answers
88 views

Eigevectors of an Idempotent Matrix: A^2 = A [Strang P310 6.2.25]

Source: P4 of http://www.minho-kim.com/courses/11sp71007/data/hw05-solution.pdf $\Large{{1.}}$ I perceive : Because $A\mathbf{a_i = 1a_i}$ for all $1 \le i \le n$, thus $1$ is an eigenvalue of $A$ ...
4
votes
3answers
67 views

Any vector is a linear combination of the eigenvectors ? [Strang P296 6.1.25]

Suppose $A$ and $B$ have the same eigenvalues $\lambda_1, \cdots, \lambda_n $ with the same independent eigenvectors $\mathbf{x_1, \cdots, x_n}$. Then $A = B$. Reason: Any vector $\mathbf{x}$ ...
4
votes
2answers
79 views

Connection between even/odd and symmetric/skew symmetric

I read awhile back that the set of continuous real valued functions from $\mathbb{R} \to \mathbb{R} $ has a direct sum decomposition into subspaces of strictly even and odd functions. Any such ...
5
votes
2answers
90 views

Orthornomal matrices [duplicate]

Is there a more direct reason for the following: If the columns of $n\times n$ square matrix are orthonormal, then its rows are also orthonormal. The standard proof involves showing that left ...
8
votes
1answer
84 views

Intuition & Proof of rank(AB) $\le$ min{rank(A), rank(B)} (without inverses or maps) [Poole P217 3.6.59, 60]

I'm aware of analogous threads; I hope that mine is specific enough not to be esteemed one. $\mathbf{a^i}$ is a row vector. $A, B$ are matrices. Prove: $1$. $\mathbf{a^i}B$ is a linear ...
5
votes
2answers
61 views

How to Intuit if these are Linear Transformations or not ? [Strang P380, 7.1.3(c), (d)]

On P376, Strang writes : "You'll get good at recognising which transformations are linear". In his video lectures, he does this; before algebra, he previses whether something's a linear transformation ...
2
votes
1answer
60 views

Intuition/Picture - Theorems on Linear Independence, Span, Basis, Dimension [Poole, Section 6.2]

I'd like to ask about the intuitions for these theorems, absent in David Poole's Linear Algebra (to which the page numbers refer). Also, are there pictures for these theorems?
11
votes
1answer
490 views

Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
2
votes
0answers
29 views

How to Intuit/See Matrix Factorisation [GStrang P250 Ex 5.1A]

I beg leave for your forgiveness over the colours. Please enlighten me if there's a more efficient way. How is the determinant of the checkerboard sign pattern matrix, $ \begin{bmatrix} a(1, ...
2
votes
1answer
29 views

Show that a $5 \times 4$ Matrix of row rank $3$ has nontrivial solutions for $Ax=0$

Statement: Let $A$ be a $5 \times 4$-Matrix with row rank 3. Show that $Ax=0$ has nontrivial solutions. This is a homework problem, as a hint I am given one formula (which we didn't discuss in ...
4
votes
0answers
58 views

Intuition - Zero Vector, Its Existence in Any Set, and Linear Dependence [GStrang, P169]

$I.$ The zero vector is linearly dependent. $II.$ Any set containing the zero vector must be linearly dependent. I only apprehend the truths of I and II above from the definition of linear ...
0
votes
2answers
134 views

Tricky question in Matrices! [closed]

Define $$A=[\text{I}+\sum_{k=1}^{m}u_{k}u_{k}^T]^{-1}$$, where for each $u_k$ is a $0-1$ column vector. Prove that for every $1\leq k \leq m$ $$Au_{k}u_{k}^T\geq0$$ i.e. each entry of $Au_ku_k^T$ ...
5
votes
2answers
73 views

Do I influence myself more than my neighbors?

Consider relations between people is defined by a weighted symmetric undirected graph $W$, and $w_{ij}$ shows amount of weight $i$ has for $j$. Assume all weights are non-negative and less than $1$ ...
3
votes
1answer
107 views

Linear (in)dependence of $\sin (x), \sin (x+1), \sin (x+2)$

I need to discuss the linear independence of the following given vectors: \begin{align} \sin(x), \sin(x+1), \sin(x+2)\end{align} there are many similar questions on math.SE but most of which I have ...
2
votes
2answers
93 views

Intuition for Geometric Transformations

I've been making a lot of effort over the past few hours to gain some intuition into the art of geometric transformation but to little avail. I would really like to be able to look at a transformation ...
1
vote
1answer
46 views

On subspace verification

I am struggling with the following Problem: \begin{align}Y= \lbrace (x^4-y^4,0,0,0) \mid x,y \in \mathbb{R} \rbrace \subset \mathbb{R}^4 \end{align} Question, is the given Set a subspace of ...
3
votes
1answer
46 views

If $A$ is the adjacency matrix of a graph, why does the $(i,j)$ entry of $A^n$ give the number of $n$-step walks from $i$th vertex to $j$th vertex?

Let $A$ be the adjacency matrix of some directed graph with $m$ vertices labeled as $v_1, v_2, \ldots, v_m$. So here $A_{ij} = 1$ if there is an edge from $v_i$ to $v_j$, and $A_{ij} = 0$ otherwise. ...
1
vote
2answers
379 views

How to think about the change-of-coordinates matrix $P_{\mathcal{C}\leftarrow\mathcal{B}}$

I've taken a linear algebra course in the past, but I feel my understanding of coordinate change is very superficial. For example this exercise (4.7.1 from Lay's "Linear Algebra and its Applications" ...
2
votes
3answers
164 views

Geometric interpretation of the addition of linear equations in general form

I have a very simple question: suppose I have two 2D linear equations in general form $$ a_1x + b_1y + c_1 = 0$$ $$ a_2x + b_2y + c_2 = 0$$ I'd like to know what's the (intuitive) geometric ...
0
votes
0answers
34 views

Intution of eigenvalues of a matrix [duplicate]

It is easy to calculate the eigenvalues and eigenvectors of a matrix in linear algebra. But what does this mean intuitively?
0
votes
1answer
43 views

What comes first, a vector base or orthogonality?

Pick any three vectors of a vector space randomly (but linearly independent). Then we assign them coordinates: $$e_1=[1 0 0]$$ $$e_2=[0 1 0]$$ $$e_3=[0 0 1]$$ Therefore now they are orthonormal ...
1
vote
1answer
94 views

Finding a mapping such that its kernel equals the image of another non bijective mapping

For an $a \in \mathbb{R}$ let $\phi_a: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear mapping such that $\phi_a(x) := \begin{pmatrix} 1 & 2 & 2 \\1 & 3 & 5 \\ 1 & -1 & a ...
30
votes
2answers
665 views

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
4
votes
2answers
146 views

Matrices Intuition

I am currently studying matrix algebra. The axioms and theorems of this form of algebra are a bit different from the high school algebra I did. However one knows that one is dealing with real numbers ...
1
vote
1answer
104 views

singleton null vector set linearly dependent, but other singletons are linearly independent set

Why the set $\{\theta_v\}$ where $\theta_v$ is the null vector of a vector space is a dependent set intuitively (what is the source of dependence) and the singleton vector set which are non-null are ...
1
vote
0answers
81 views

A basic intuitive question on basis

From Zorn's lemma, basis can be thought of as a maximal independent set as well as minimum cover (covering all the vectors). Is this observation correct ? Can this observation be related to the usual ...
0
votes
3answers
78 views

A basic doubt on linear dependence and basis vectors

I see that linear independence/dependence is defined for a finite set of vectors in books. But, basis vectors are always independent and they need not be finite. Is the definition consistent ?
2
votes
2answers
214 views

Intuition/Understanding of Inverse and Determinants

This is not homework, but extends from a proof in my book. EDIT We're given an $m \times m$ nonsingular matrix $B$. According to the definition of an inverse, we can calculate each element of a ...
2
votes
3answers
373 views

Calculating limits using the $\epsilon$-$\delta$ definition.

Suppose you have a function $f(x)=( x^2-4)/(x-2)$. How then do we find the limit as $x\to2$ in accordance with the epsilon delta definition? I mean suppose we don't know how to calculate limit and we ...
5
votes
2answers
169 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
1
vote
0answers
96 views

What really are determinants? [closed]

As a layman, it is not clear to me what does a determinant stand for and how it could be computed. So, what really is a determinant, meaning, why should we care about them and how is the best way to ...
1
vote
2answers
127 views

Confused with Eigenvalues and Eigenvectors and Vector transformations

Hello fellow mathematicians, I am studding " Eigenvalues and Eigenvectors " at this point and I need to understand something here: I am actually performing automatic operations on finding them, but ...
0
votes
2answers
100 views

Intuition for orthogonal vectors in $\Bbb R^n$

Two vectors in $\Bbb R^n$ are orthogonal iff their dot product is $0$. I'm aware that the dot product can be defined in other spaces, but to keep things simple let's restrict ourselves to $\Bbb R^n$. ...
1
vote
1answer
41 views

Why is $E_{\lambda}$ the kernel of the linear map $\alpha-\lambda I$

The book starts the chapter on Eigenvalues and Eigenvectors, and goes that this statement is obvious. Here $E_{\lambda}$ stands for the set of vectors $v$ such that $α(v) = λv$, for any scalar ...
0
votes
1answer
547 views

The orthogonal projection onto a plane - explanation

Could somebody explain, why orthogonal projection onto a plane with equation $x_1+x_2+x_3=0$ is given by $$y=(x_1,x_2,x_3)-\bigg( \frac{x_1+x_2+x_3}{3}\bigg)(1,1,1)$$ I don't understand, why we sum ...
0
votes
1answer
93 views

cauchy schwarz equality: difference in proving style for linear algebra and expectation version

I am interested in proving the following sub version of Cauchy Schawrz equality. 1) LA version : If $x$ and $y$ are two real vectors and the following holds $$<x,y> = ||x||.||y||$$ then $x$ ...
15
votes
5answers
568 views

An intuitive approach to the Jordan Normal form.

I want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a Mathematician. As far as I understand this the idea is to get the closest representation of an ...