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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dimension of quotient space

Let $f(x)=x^4+3x^3-x^2-4x-3$ and $g(x)=3x^3+10x^2+2x-3$ and $U = \{u(x)f(x)+v(x)g(x) | u(x),v(x) \in \mathbb{F}[x]\}$, find the dimension of quotient space $\mathbb{F}[x]/U$ If $V$ is a finite ...
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66 views

Solving for first term in vector product

I'm trying to solve a system of equations for a physics application I've been working on, and I'm down to one thing left that's stumping me. Essentially, I need to solve $$A \times B = X$$ where $A, ...
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2answers
120 views

Squaring a basis = basis? [closed]

Let $b_1, \dots, b_n$ be vectors in $\mathbb R^n$ with positive entries, i.e. $b_1, \dots, b_n \in \mathbb R_{>0}^n$. Is it then true that the following vectors $$ \begin{pmatrix} (b_1^{(1)})^2 ...
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0answers
52 views

Lagrange multiplier for more than one constraints.

How to minimize $x^TAx$ over the set $D=(x\geq 0, x^TBx=1$ and $(I-A^\dagger A)x=0$), where $A$ is copositive matrix of order $n-1$ and $B$ is strictly copositive matrix of order $n$. If I drop the ...
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3answers
176 views

condition on the dimensions of a matrix for its inverse to exist

Let $A$ be an $m\times n$ matrix, what condition on the dimensions $m$ and $n$ is necessary for the quantity $(A^t\times A)^{-1}$ to exist? Please kindly provide your explanation. Many thanks!
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3k views

Show that the equation Ax=x can be rewritten as (A-I)x = 0 and use this result to solve Ax=x for x.

Given matrix A = \begin{bmatrix}2 & 1 & 2 \\ 2 & 2 & -2 \\ 3 & 1 & 1\end{bmatrix} and x = \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix} Answer: Given Ax = x. Subtract from ...
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135 views

Why does elementary row operation on combination of Identity matrix and invertible matrix yields inverse matrix?

Why does elementary row operation on combination of Identity matrix and invertible matrix yields inverse matrix? $$A = \begin{pmatrix} 1 & 3\\ 4 & 2 \end{pmatrix}$$ if we combine $A$ with ...
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1answer
158 views

composition sum of functions/sum of composition of functions

I know it sounds really dumb, but is it true that $(f_1+f_2)\circ g=f_1\circ g+f_2\circ g$? I know it must be really elementary, but I don't recall seeing this being proved (or defined) explicitly.
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1answer
36 views

Triples with even intersection

Let $\mathfrak M=\{M_1, \ldots , M_s\}$ be a collection of triples of natural numbers from $1$ to $n$, such that $|M_i \cap M_j| \ne 1$ (or, equally, $|M_i \cap M_j|$ even for $i \ne j$. How large can ...
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1answer
47 views

Basic matrix question

Given $A$ is a square matrix satisfying $A^2=A$, and $B$ any matrix of the same size as $A$, show that $$(AB-ABA)^2=0.$$ Tried to expand and solve, and also tried to show $AB=ABA$ with the ...
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35 views

When are you able to reduce equations such as $\tan(\pi/2-2x)=\tan3x$ to simply $\pi/2-2x=3x$?

as the title says, I am unsure when I can do this. Does this only apply to specific trigonometric functions? Any help clarifying this would be appreciated.
3
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1answer
41 views

Why is this statement about $\text{Span}$ false?

Here is a true-false question known to be false: If $\mathbf{a}$ is in $\text{Span} \left \{ \mathbf{b}, \mathbf{c} \right \}$, then $\mathbf{b}$ is in ...
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1answer
277 views

Using the pseudoinverse to find the linear combination of functions?

I'm working out this problem with a friend of mine on a group project and we are both stuck Our professor insists that we do all of our work in Maple. I like Maple, but it's not as great as ...
0
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1answer
47 views

showing functions are linearly independent subsets

Let $n$ be a positive integer and let $V_n$ be the vector space of polynomial functions (in $t$) from $F$ to $F$, where $F$ is some field. Suppose also that $F$ has more than $n$ elements. Let $f_0, ...
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1answer
45 views

Properties determining boundedness of function

The function I am looking at is $$f(x) = \frac{1}{2}x^TAx + b^Tx + c$$ where $A$ is a symmetric matrix in $\mathbb{R}^{n\times n}$ and $b,c$ belong to $\mathbb{R}^n$ I want to determine what ...
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1answer
112 views

Finding a basis of the kernel

I'm having a bit of difficulty finding the basis of kernel given a matrix. The matrix is a 2x4 matrix with entries (-9 -3 -3 3) (-9 -3 -3 3) which I then put into rref as (1 1/3 1/3 -1/3) (0 0 0 0). ...
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1answer
2k views

What does “IR” mean in linear algebra?

I am new to linear algebra and I was wondering if I could get some help for this question. I understand if it was something like this IR^2 -> IR. I have no idea what IR(=IR^1) means. Could someone ...
3
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1answer
72 views

Matrix question: implication of $\frac{1}{n}X'X\to M$

Suppose $K$ is fixed and consider a matrix $X$ that is $n\times K$ and has full column rank. Assume that we know $$ \frac{1}{n}X'X\to M\text{ as } n\to\infty.\tag{i} $$ That is, as $n$ becomes larger, ...
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0answers
45 views

coordinates of ordered basis

Suppose $n>2$ is an integer. Consider the ordered basis $\{B_0,B_1,...,B_n\}$ of the space of polynomials over the complex numbers of degree $n$ or less, call it $P_n$ where $B_k=t^k$. Consider ...
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1answer
64 views

Quadratic form's transition matrix

Quadratic forms's matrix is $$A=\begin{bmatrix} 9 & -2 \\ -2 & 6 \end{bmatrix}$$ First, I have to find it's matrix in relation to some canonical basis. After applying elementary ...
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1answer
89 views

Help on the relationship of a basis and a dual basis

If $B_1 = \{v_1,\ldots,v_n\}$ and $B_2 = \{v_1',\ldots,v_n'\}$ are basis for a vector space V, and $D_1= \{\delta v_1,\ldots, \delta v_n\}$ and $D_2 = \{\delta v_1',\ldots, \delta v_n'\}$ are the ...
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1answer
195 views

I want to find a spectral decomposition of the of the matrix $B$ given the following information.

I want to find the spectral decomposition of the of the matrix $B$ given the following information: A =\begin{pmatrix} 2 & 1\\ 1 & 2\\ \end{pmatrix} with $c \gt 0$ and $B = cA$. I found ...
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0answers
43 views

Can a mapping be from 1 to many?

I'm starting to learn linear algebra from Lang's Linear Algebra, and am a little confused about the definition of mapping. A mapping is defined to be an association which to every element of $S$ ...
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1answer
69 views

Find a corresponding orthonormal set

I was able to show that the functions 1, cos(nx), sin(nx), n = 1,2,3,... form an orthogonal set for the inner product $$(f,g)=\int_{-\pi}^\pi f(x)\overline {g(x)}dx$$ on the space of continuous ...
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1answer
169 views

Problem on matrices : $\dim E\leq n^2-(n-r)^2-1$

I have the following problem : Let $E$ be a subspace of $M_n(\mathbb{R})$ that contains no invertible matrix. Let $r=\max\{rank(M)\mid M\in E\}$ Show that $\dim E\leq n^2-(n-r)^2-1$ I don't know ...
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1answer
25 views

Find a change of coordinates matrix

Find the change of coordinates matrix that changes coordinates in the basis $1$, $1+t$ in $P_1$ to the coordinates in $1-t$, $2t$.
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59 views

About the dimension of a subspace of a vector space of linear transformations

Suppose that $X$ and $Y$ are vector subspaces of $V$ and $W$ respectively. It is known that $$T = \{\alpha \in L(V, W) \mid α(x) \in Y \text{ for all } x \in X\}$$ is a vector subspace of $L(V, W)$. ...
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1answer
92 views

Bases and dimensions of two spans

Let $$U=\operatorname{span}\{(1,1,0,-1),(1,2,3,0),(2,3,3,-1)\}$$ and $$ W=\operatorname{span}\{(1,2,2,-2),(2,3,2,-3),(1,3,4,-3)\}$$ I am supposed to find basis and dimensions for $U+W$ and $U\cap W$. ...
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1answer
138 views

Quotient Space (W-Affine Subspaces) “Proof” Verification.

Let $(V, K)$ be a vector space and $W ⊂ V$ a subspace. A subset $S ⊂ V$ is called a $W$-affine subspace of $V$ if the following holds $∀s, s ∈ S, s − s ∈ W$ and $∀s ∈ S, ∀w ∈ W, s + w ∈ S$. ...
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3answers
31 views

Prove orthogonality in C

I know that in $\mathbb{R}^n$ $\parallel x^2 + y^2\parallel \ =\ \parallel x\parallel^2 + \parallel y \parallel^2$ IFF $x \perp y$. What I don't understand is why this is false for $\mathbb{C}^n$. ...
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2answers
167 views

Matrix characterization of surjective and injective linear functions

I don't remember well of my Linear Algebra classes, looking the rank of a matrix $A\in M(n\times m)$ how can we say the application associated to this matrix is surjective or injective? For a matrix ...
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1answer
53 views

The relation between two different definitions of Affine sets

I am following a presentation, which says that for an affine set $L \subseteq \mathbb{R}^n$ it is: $$L=\left\{x|Ax=b \right\}$$ for some $A,b$. The first definition of $L$ as an affine set is given ...
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1answer
510 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
2
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1answer
106 views

Proving the properties of subspace (coset)

Let $W$ be a subspace of a vector space $V$ over a field $F$. Let $v\in V$. We define $$v+W=\{{v+w:\ w\in W\}}$$ Show that i) $v+W$ is a subspace of $V$ if and only if $v\in W$ ii) ...
0
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1answer
43 views

To calculate the order of the group of all invertible linear operators on a vector space

Let $V$ be a finite dimensional vector space over a finite field $F$ , then how can we calculate the order of the group of all invertible linear operators on $V$ in terms of $|F|$ and $\dim V$ ?
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2answers
55 views

Let $\theta : V \rightarrow V$ be an idempotent linear map. $W = \text{ker }\theta$. if $\theta$ not identity map, prove $W \neq \{0_v\}$

Let $\theta : V \rightarrow V$ be an idempotent linear map. $W = \text{ker }\theta$. if $\theta$ not identity map, prove $W \neq \{0_v\}$ I think I might have solved it, but I just need my proof ...
2
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4answers
255 views

how inner products are defined on a vector space?

How do mathematicians define inner product on a vector space. For example: $a = (x_1,x_2)$ & $ b =(y_1,y_2) $ in $ \mathbb{R}^2.$ Define $\langle a,b\rangle= x_1y_1-x_2y_1-x_1y_2+4x_2y_2$. ...
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2answers
51 views

Let $V$ be finite dimensional v.s. and $0 \ne T\in \mathscr L(V)$ , then $\exists$ $S \in \mathscr L(V)$ such that $0 \ne T \circ S$ is idempotent

If $V$ is a finite dimensional vector space and $T \ne0$ is a linear operator on $V$ , then how may we prove that there is a linear operator $S$ on $V$ such that $T\circ S$ is non-zero and idempotent? ...
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255 views

How do I test if a set of matrices is a subspace?

I have some subsets of matrices defined for me, and I want to test if those are a subspace. I know that the definition says that if: $x, y \in M \Rightarrow x+y \in M$ $x \in M, \lambda \in ...
6
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2answers
270 views

Unique solution comes out to be trace

Let $f: \mathcal{M}_n(F)\to F$ be a linear functional satisfying $f(\mathbf{AB})=f(\mathbf{BA})\;\forall \mathbf{A,B}\in \mathcal{M}_n(F)$. Also it satisfies $f(\mathbf{I})=n$. Prove that ...
0
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1answer
59 views

Eigenvalues & eigenvectors of a matrix [duplicate]

I have a couple of questions regarding eigenvalues and eigenvectors. Let $A=\begin{pmatrix}4 & 2 \\ 5 & 1\end{pmatrix}$, $\mathbf{u}=\begin{pmatrix}2\\-5\end{pmatrix},\mathbf{v}=-2\mathbf{u}$ ...
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2answers
36 views

Cyclic space and commuting linear transforms

Suppose that $V$ is a $n$ dimensional vector space, $\sigma$, $\tau$ are two linear transformation on it. Suppose further that $V$ is a cyclic space of $\sigma$, that is, there exists a $\alpha\in V$ ...
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4answers
798 views

Exercise books in linear algebra and geometry

I'm studying Brannan's Geometry and Lang's Introduction to Linear Algebra and I was wondering if there are some exercise books (that is, books with solved problems and exercises) that I can use as ...
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3answers
300 views

How to prove that every linear operator on a finite dimensional vector space is a sum of invertible linear operators

Let $V$ be a finite dimensional vector space , then how do we prove that for every linear operator $T$ on $V$ , there exist invertible linear operators $S_T' , S_T'',...$ such that $T(\vec v)=S_T' ...
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1answer
39 views

$A$ is $n\times n$ real matrix with $A^2=-I$ , to prove that $n$ is even

Let $A$ be a $n\times n$ matrix with real entries such that $A^2=-I$ , then how do we prove that $n$ is even ? ( All I know about this matrix is that it has no real eigenvalues )
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3answers
59 views

Counterexamples of the statement

Find counterexamples to the following statements: (1) In every field $F$, if $a \in F$, $a + a = 0$, then $a = 0$; (2) In every field$ F, −1 \neq 1$
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1answer
63 views

What is the (algebraic) dimension of the dual of a vector space?

Let $V$ be a vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what is the dimension of ...
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0answers
83 views

Check my proof - Linear Algebra

Still not completely confident with my capabilities in writing formal proofs so I thought I would ask for a check of this proof. Theorem Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be ...
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2answers
50 views

How do you define Dimensions in general?

I am having a hard time understanding the definitions of dimensions. Dimension of a finite-dimensional vector space is defined as the length of any basis of the vector space. The definition seems to ...
1
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2answers
280 views

Geometric intuition for linear independence

My textbook states: If a subspace $V$ of $\mathbb{R^3}$ is a plane, our geometric intuition tells us that we can find at most two linearly independent vectors in $V$, and we need at least two ...