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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2answers
118 views

Proof that this is independent

Prove that {$1, \sin(x), \sin(2x), \sin(3x),\ldots, \sin(nx)$} is an independent set. I can think of the long way which is to differentiate this and put the differentiations into a matrix and row ...
3
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1answer
231 views

Finding matrix norm equivalence constants

I've been given the following: "Find the best positive constants $\alpha$ and $\beta$ such that $\alpha\left\|A\right\|_2\leq\left\|A\right\|_1\leq\beta\left\|A\right\|_2$ for all ...
0
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1answer
93 views

How to prove that the negative of any $\vec{v}\in V$ is unique?

I can't help but come up with a solution that feels wrong. I am using a fact to prove something, when I am asked to prove that fact... $\vec{v}$ is a negative of $\vec{w}$ if ...
3
votes
1answer
188 views

Closed-form expressions for dual norms of real normed vector spaces

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$. The "dual ...
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0answers
73 views

Invertible Composition of Maps

How can I show that if a composition of two linear maps is invertible, the two maps must also be invertible?
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2answers
1k views

Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$?

W is a subspace of $\mathbb{R}^n$ iff The zero vector ∈ W. X + Y ∈ W for any X, Y ∈ W. aX ∈ W for any X ∈ W and a ∈ R. So, given W = { X : X = [x1...], x1 = 0, x2 = 0, ... xn = 0 } ∈ Rn The zero ...
0
votes
1answer
39 views

Time of point colliding with a moving line

In 2-D space, given a line defined by two points a and b, and a third point c that is not initially (t=0) in the line defined by a and b, is it possible to obtain an expression for the numerically ...
3
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1answer
377 views

If $A^2=-I$, Prove that $\det{A}=1$

If $A^2=-I$ , where $A$ is a square matrix of order $n$ and which contains real entries only and $I$ is identity matrix. Then how can we prove that $\det(A)=1$?. I could prove that $n$ should be an ...
2
votes
3answers
117 views

Solving a system of three linear equations with three unknowns

Is my working correct or am I completely wrong? Have I missed anything out? Any feedback is appreciated. Question: Consider the following system of equations $2x + 2y + z = 2$ $−x + 2y − z = −5$ ...
3
votes
1answer
375 views

Column and Row Picture for Singular System of 100 Equations (Strang P55, 2.2.32)

Start with 100 equations $\color{#8F00FF}{A}\mathbf{x} = \mathbf{0}$ for $\mathbf{x} = (x_1, ..., x_{1oo})$. Suppose elimination reduces the 100th equation to $0 = 0$, so the system is "singular". ...
1
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1answer
70 views

Showing a vectorspace equals a span of polynomials?

I'm not sure on the titleing of this question, mostly because I don't really understand the question this task is asking me. Can someone point me in the right direction? Let $H_{2,3}\subset P_{2,3}$ ...
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1answer
85 views

Statement: For every real number $x$, if $x^4 + 4x^2 - 4x$ is less than zero, then x must be a number between 0 and 1

I just want to double check if I got this question correct. Statement: For every real number $x$, if $x^4 + 4x^2 - 4x$ is less than zero, then x must be a number between $0$ and $1$ a) Rewrite the ...
0
votes
2answers
131 views

Prove that for any real or complex scalar $\lambda$ the set $S$ is a subspace of $\mathbb{R} ^n$ or $\mathbb{C} ^n$

Let $A$ be an $n \times n$ real or complex matrix . Prove that for any real or complex scalar $\lambda$ the set $S = \{x : Ax = \lambda x \}$ is a subspace of $\mathbb{R} ^n$ or $\mathbb{C} ^n$. So ...
0
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2answers
124 views

Why determinant map matrices is a polynomial and not identically zero?

Let $A,B \in M_n(C)$ are invertible then we consider the map $c \rightarrow det(A+cB)$ which is a polynomial. How to prove that the polynomial $det(A+cB)$ not identically zero? thanks in advanced.
1
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1answer
449 views

Relationship between eigenvalues and eigenvectors of square invertible A and its inverse?

So what can we say about the relationship between eigenvalues and eigenvectors of square invertible $A$ and its inverse $A^{-1}$? We know that $A$ is invertible iff all its eigenvalues are nonzero, ...
1
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2answers
58 views

$P+cQ$ is invertible for a finite number

Since $C$ is a field and $P,Q \in M_n(C)$ are invertible, can any body show me that $P+cQ$ is invertible for all but a finite number $c \in C$
0
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1answer
51 views

Question about specific linear operator that does not have inverse

Consider $\mathbb{R}[x]$ = the set of real polynomials, and let f(x) = d/dx. Then what is g(x) so that f(g(x)) is identity (but g(f(x)) is not)? Sorr my calculus is a little rusty. I considered ...
0
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1answer
258 views

Sylvester's criterion about positive definite matrices.

The below quote is copy from "Problems and Theorems in Linear Algebra" Author is : V.Prasolov Let $A=||a_{ij}||_{1}^{n}$ be an Hermitian matrix, if $A$ is positive definite, then the matrix ...
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vote
2answers
579 views

Write the solution set as a span of four vectors

Write the solution set of $$2x+3y-3z+w+v=0$$ as a span of four vectors (i.e. find four vectors in $\mathbb{R}^5$ so that their span in $\mathbb{R}^5$ is the solution set of this equation). I'm ...
0
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0answers
80 views

Need to find components of a vector given distance and angle from a known vector on a known plane

I am looking for a way to compute the components of a vector (C in illustration) given: components of A and B, the angle between B and C, and the magnitude of C. I am looking for a way to solve this ...
1
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0answers
82 views

A lattice-theoretic question related to noncommutative tori

[NCG] So I'm trying to pin down a fairly well-known bit of noncommutative-geometric folklore that says that for $\Theta \in M_N(\mathbb{Q})$ skew-symmetric, the corresponding noncommutative $N$-torus ...
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2answers
1k views

Prove that the determinant of a householder matrix is -1

I understand that a householder matrix has eigenvalues of either 1 or -1, however I isn't clear to me why the determinant is -1. Clearly the determinant is equal to the product of the eigenvalues so ...
2
votes
1answer
35 views

What is the notation $M_{n}(\mathbb{R})$?

I'm familiar with $M_{m\times n}(\mathbb{R})$ being the set of all $m\times n$ matrices, but I'm not sure I know what this one is.
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0answers
48 views

How is open set defined in linear map space

I got this statement in my homework: Prove that the invertible linear contractions are an open set in $Mat(2\times 2;\mathbb{R})$ I know what "invertible linear contraction" , "open set" and ...
0
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2answers
57 views

percentage problem in solution

I have solve this percentage problem by back solving method. Can't I solve this straight forwardly and easy way?
1
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2answers
176 views

Orthogonal projection and subspaces proof?

Let's let $M$ be a subspace of $\mathbb{R}^n$ and let $N$ be a subspace of $M$. Let $m$ and $n$ denote the orthogonal projection matrices onto $M$ and $N$. Show that $mn = nm = n$. -- I'm thinking ...
0
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2answers
45 views

Using $y = (n / 0)(x - t)$ for the equation of a vertical line

Can anyone tell me if this would be a valid equation for a vertical line? y = $(n / 0)(x - t)$ for $n$ = all real numbers $\ne 0$ and $t =$ x-intercept I've tested it to the best of my ...
1
vote
3answers
4k views

Finding basis for the space spanned by some vectors.

Find a subset of vectors $\{v_1, v_2, v_3, v_4, v_5\}$ that forms the basis for the space spanned by these vectors: $$v1=\left ( \begin{array}{c} 1\\-2\\0\\3 \end{array}\right), v2= \left ( ...
0
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1answer
46 views

More trouble with the same proof

I asked a question about a proof and now I have two more questions. Here is how I understand the proof (second half): If $m_T$ has no multiple roots it looks like this: ...
2
votes
1answer
72 views

Show that $\max_i\left\{|v_i| + |w_i|\right\} \leq \max_i\left\{|v_i|\right\} + \max_i\left\{|w_i|\right\}$.

While trying to prove that the $\infty$-norm of a vector in $\mathbb{R}^n$ does satisfy the properties of being a norm, I inevitably came across the following inequality: $\max_i\left\{|v_i| + ...
1
vote
1answer
43 views

basis for $\mathbb{R}^{\{0, 1\}^S}$

Let's say $S$ is a finite set. If $R$ is any subset, then consider $f_R$ defined by $f_R(\eta)=\prod_{r \in R} (2\eta(r)-1)$ whenever $\eta \in X:= \{0, 1\}^S$. Why is this a real basis for the ...
0
votes
1answer
20 views

Difference operator endomorphism

Let $\delta : R_{p}[x] \to R_{p}[X] $ the endomorphism of $R_{p}[X]$ such that : $\delta(P(X)) = P(X + 1) - P(X)$ , what is the kernel of $\delta$ ? (i tried to compute it explicitly but that was a ...
0
votes
1answer
468 views

Show that any complex roots must occur in conjugate pairs. [duplicate]

Let $p(\lambda)=c_0 + c_1\lambda + \dots + c_n\lambda^n$ be a polynomial with real coefficients. Show that any complex roots of $p(\lambda)=0$ must occur in conjugate pairs, i.e. if ...
0
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2answers
152 views

Either prove or disprove the given statement, For all x∈ℝ, there exists y∈ℝ such that (x^2 + y = 1)

Can someone please help me understand this question? I'm sort of confused. Express the negation of the statement without using the word "not" or the ¬ symbol. Then either prove or disprove the ...
1
vote
2answers
48 views

Prove linear dependency in $\mathbb{F}_{m}[z]$

Let $\mathbb{F}_{m}[z]$ denote the vector space of all polynomials with degree less than or equal to $m\in\mathbb{Z}_{+}$ and having coefficients over $\mathbb{F}$, and suppose that ...
0
votes
1answer
34 views

One way to describe the pattern of covariation for a linear function is:

One way to describe the pattern of covariation for a linear function is: As input value increases by 1, the output value changes by a constant (fixed) amount k where k is some real number. Explain why ...
3
votes
1answer
128 views

Characteristic Polynomial of $A$ and polynomials annihilating $A$

If $A$ is a real $3 \times 3$ matrix which is not diagonal. $p$ is a polynomial of degree 3 with real coefficients which is annihilating $A$. I have proved that if $A$ has a complex root (with non ...
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2answers
28 views

Having trouble with one step of computation

I am solving a few exercise I found online. My trouble is with solution 5.b) the second direction. Where it is stated that it follows that $(T_m - \lambda_j)$ is invertible on $U_m$ for $j = ...
0
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2answers
47 views

A list of length $1$.

I need to verify that list $v$ of length $1$ is linearly independent if and only if $v \neq 0$. I just need to confirm that I am correct in the following reasoning: Let $\text{span}() = \{0\}$. Now ...
6
votes
1answer
144 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
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3answers
176 views

How can we solve this elementary school math problem without using equation or simultaneous equations?

I got an elementary math problem this morning. The students have not learnt linear equation as well as simultaneous equations yet. I have solved it with simultaneous equations and I have no idea ...
3
votes
6answers
4k views

Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue.

I can't find the missing link between singularity and zero eigenvalues as is stated in the following proposition: A matrix $A$ is singular if and only if $0$ is an eigenvalue. Could anyone shed some ...
4
votes
0answers
52 views

Decomposability in the tensor product sense of functions of two variables

Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$. Let $F$ be a function $$ F:S\times T\longrightarrow K $$ where $K$ is a topological field ...
0
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1answer
46 views

Conditions for a system to be solvable.

I have the following system of equations: $$\begin{aligned} \left\{\begin{array}{l} a+dz+cy+exy = 0\\ 10a+3bx-exy =0\\ -5a-dz = 0 \end{array}\right. \end{aligned}~~.$$ I would like to solve for ...
1
vote
1answer
441 views

how do I find the distance between a vector and a span of a set of vectors?

I have a set of vectors $A = \{ v_1, \ldots, v_n \}$ and an additional vector $w$ all in $\mathbb{R}^d$. I want an algorithm that finds the distance between $w$ and the span of $A$. What would be an ...
1
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0answers
119 views

A Rank-One Reduction Formula

Consider $A_{m\times n}$ is very large, dense and full rank matrix. How can I find matrix B such that $\operatorname{rank}(B)=\operatorname{rank}(A)-1$? (Rank reduction formula must be invertible and ...
6
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1answer
931 views

Row and Column Picture of a 3 x 3 Singular Matrix (Strang P43, 2.1.32)

Suppose $\mathbf{u}$ and $\mathbf{v}$ are the first two columns of a 3 by 3 matrix $A$. Which third columns $\mathbf{w}$ would make this matrix singular? Describe a typical column picture of ...
1
vote
1answer
57 views

Three numbers, one of the number's digit sum is equal to two other digit difference

So as the title says I need three numbers witch has this quality : one of the numbers digit sum is equal to other two number differnce e.g. I 68 II 52 III 97 third number digit sum is 16 and its ...
0
votes
1answer
90 views

Help show/prove linear transformation

I need some help to understand how to prove the two following tasks. Also, I'm having trouble seeing the big difference between the two. 1) Let $n > 0$, and let $L_{n,k}$ be a subspace spanned by ...
2
votes
1answer
127 views

linear transformation's geometric meaning

Linear transformations can be represented using matrix, like $$v = Au$$, which transforms vector $u$ into $v$. And my intuitive understanding about linear transformations is that, it rotates the ...