0
votes
1answer
44 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 = ...
0
votes
0answers
28 views

Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P95]

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 picture, and no proofs please. $1.$ Intuitively, in ...
1
vote
2answers
64 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
4
votes
4answers
130 views

rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
1
vote
1answer
35 views

Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
2
votes
2answers
61 views

Gaining Linear Algebra Intuition — Subspaces

So I aced linear algebra over the fall semester, though I'm deeply troubled in that I struggle to really describe what I did. I cannot say with confidence what it all meant, nor do I have any sort of ...
3
votes
1answer
63 views

Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
2
votes
1answer
87 views

Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
2
votes
2answers
58 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
36 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
57 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
301 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
210 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
70 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 $\mathbb{R^3}$ ...
3
votes
1answer
48 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
49 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
54 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
2answers
427 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
26 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
123 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
136 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
88 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
97 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
100 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
68 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
63 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?
12
votes
1answer
767 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
34 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
31 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
63 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
159 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
75 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
116 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
99 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
47 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
54 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
704 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
177 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
45 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
106 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 ...
33
votes
2answers
875 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
166 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
126 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
83 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
80 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
275 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 ...
3
votes
3answers
487 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
196 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
103 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 ...