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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Linear Algebra Done Right Example 1.7

In the book Linear Algebra Done Right I came across this example for the sum of vector spaces. Did he say the second $W + U$ is still equal to 1.7 because it doesn't matter whether you say $(x+y, ...
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308 views

Show that 1 and -1 are the only eigenvalues of this linear transformation

Define $T: M_{n\times n}\to M_{n\times n}$ by $T(A):= A^t$. Note that $T$ is a linear transformation. Show that $1$ and $-1$ are the only eigenvalues of $T$. Let $\lambda$ denote an eigenvalue ...
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134 views

Why negative of a negative number is positive?

I am intrigued in seeking the philosophy underlying it. When I was trying to prove it mathematically, I was failed but later I started to analyze vectors that what they are? As an outgrowth of ...
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163 views

Proving that $\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let $A$ be real square matrix. If $\det (A^2 - I) < 0$, then $A$ has an eigenvalue $\lambda \in (-1,1)$. How to prove this?
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363 views

Do every $3$ linearly independent vectors span all of $\mathbb{R}^3$?

I am given $3$ vectors that are linearly independent. I am trying to figure our if they span all of $\mathbb{R}^3$ to declare them as basis.
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How to find an intersection of a 2 vector subspace?

Assuming we have 2 subspaces, $\mathbb W$ and $\mathbb U$ of $\mathbb V$. how to get thier intersection?
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Computing the determinant of $\operatorname{id}+aa^t$

What is an easy way to see that $\det(\operatorname{id}_n+aa^t)=1+|a|^2$ for $a\in \mathbb{R}^n$ ?
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1k views

Perron-Frobenius theorem

In the proof of the Perron-Frobenius theorem why can we take a strictly positive eigenvector corresponding to the eigenvalue $1$? Before that, why can we even take a non-negative eigenvector? Books ...
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707 views

Compactness of the set of all unitary matrices in $M_2(\mathbb{C})$

Is the set of all unitary matrices in $M_2(\mathbb{C})$ is compact? I can show that as determinant map is continuous so unitary matrices are closed but how to show they are bounded? Please help.
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262 views

eigenvector proof

Let $V$ be a vector space and $T : V\to V$ a linear transformation with the property that $T(W)\subset W$ for every subspace $W$ of $V$. How can we prove that there is an element $\lambda$ in the ...
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59 views

Show that $T(p)=p+p'$ is invertible

We have a linear transformation $T:P\rightarrow P$ where $T(p)=p+p'$ and $P$ is the vector space of all real polynomials. I want to show that $T$ invertible. I was able to prove that its ...
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why does the reduced row echelon form have the same null space as the original matrix?

What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?
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746 views

Can systems of 3 linear equations with 3 unknowns have more than one solution?

In each part,determine whether the given vector is a solution of the linear system \begin{align} 2x-4y-z&=1\\ x-3y+z&=1\\ 3x-5y-3z&=1 \end{align} (a) $(3,1,1)$ (b) $(3,-1,1)$ (c) ...
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158 views

Why is $(AB)^{-1}=A^{-1}B^{-1}?$ [closed]

If we have two matrices $A$ and $B$ then the following property is true. $$(AB)^{-1}=A^{-1}B^{-1}.$$ I can't understand how the property is true. Can anyone give me a intuitive proof for the ...
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512 views

Proving determinant using properties of determinants

$$\begin{vmatrix} 1 & 1 & 1\\ a & b & c\\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)$$ we have to solve this by using the properties of determinants without ...
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453 views

Ways Of Matrix Multiplication

Let $A \in F^{11 \times10}$ and $B\in$ $F^{10\times11}$ We only know $2$ rows of $A$ and $3$ columns of $B$. How many entries of $B\cdot A$ can we know? I think the answer is none, because there are ...
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2k views

Sum of invertible matrices proof

If we have two square matrices, $A$ and $B$. Assume that $A + B$ is invertible. Would that mean that $A^{-1} + B^{-1}$ is invertible too?
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Proving a structure is a field?

Please help with what I am doing wrong here. It has been awhile since Ive been in school and need some help. The question is: Let $F$ be a field and let $G=F\times F$. Define operations of addition ...
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433 views

Linear algebra, power of matrices

$P^{-1}AP = \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix} $ with $P= ...
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520 views

Prove the matrix satisfies the equation $A^2 -4A-5I=0$ [closed]

How to prove that $$ A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $$ satisfies the equation $A^2 -4A-5I=0$?
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124 views

Prove or disprove: If $A^2$ is normal matrix then $A$ is normal matrix

Prove or disprove: If $A^2$ is normal matrix then $A$ is normal matrix. I think this is wrong but simply can't build a counterexample. Any hints on how to build a counterexample? There are many ...
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94 views

Linear combination $\{2,3\}$

I am trying to write \begin{bmatrix} 2 \\[0.3em] 3 \end{bmatrix} as a linear combination of \begin{bmatrix} 1 \\[0.3em] -1 \end{bmatrix} ...
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411 views

Similar matrices properties

So I have a question which I can not solve. Assuming $A,B \in \mathbb{M_{n}(\mathbb{R})}$, $A$ similar to $B$, is it possible that $\det(A) = \det(B^{2})+1$? We know that there exists $P$ ...
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Linear Transformation from $ \mathbb R^2 \rightarrow \mathbb R^2 $

Let $ v_1 = \begin{bmatrix} 1 \\ -1 \\ \end{bmatrix} $ and $ v_2 = \begin{bmatrix} 2 \\ -3 \\ \end{bmatrix} $ Let $ \mathbb R^2 \rightarrow \mathbb R^2 $ be linear transformation satisfying $ ...
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544 views

does multiplication of singular matrix with some matrix result in singular matrix?

Suppose that there is singular (non-invertible) matrix $A$. If it gets multiplied by any square matrix $B$, would $AB$ be singular?
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Kernel of Linear Functionals

Problem: Prove that for all non zero linear functionials $f:M\to\mathbb{K}$ where $M$ is a vector space over field $\mathbb{K}$, subspace $(f^{-1}(0))$ is of co-dimension one. Could someone solve ...
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1k views

What is $\mathbb Z[[t]]$? What are the double brackets?

What does $\mathbb{Z}[[t]]$ mean? Why are there double square brackets? I can't search through Google, because I can't search Latex.
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251 views

About finite dimensional vector spaces

Would you tell me why the statement below holds? A vector space $V$ has a basis if and only if $0 < \dim V < \infty.$
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How to prove the uniqueness of the solution of $ax+b=0$?

I have no background in mathematical analysis or the like, but I am interested to know how to prove the uniqueness of the solution of $ax+b=0$? Perhaps your answers will help me to prove other ...
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148 views

Every automorphism of $\mathbb{R}^n$ a linear mapping

Is there an automorphism of $\mathbb{R}^n$ (here it is seen as a vector space) that is not a linear mapping?
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109 views

How is $\text{tr}(AB) = \text{tr}(BA)$?

I have already seen theoretical proofs on how $\text{tr}(AB) =\text{tr}(BA)$, but I'd like to see how this is true concretely. Let's say I have two $2\times 2$ matrices: $A = \begin{bmatrix} ...
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60 views

If $A,B \in \Bbb R ^{n \times n}$ have same $n$ linearly independent eigenvectors, $AB=BA$.

If $A,B \in \Bbb R ^{n \times n}$ have same $n$ linearly independent eigenvectors, $AB=BA$. I know that $A,B$ are similar to the same diagonal matrix so they are similar, but how is that help to ...
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132 views

Can we prove $BA=E$ from $AB=E$? [duplicate]

I was wondering if $AB=E$ ($E$ is identity) is enough to claim $A^{-1} = B$ or if we also need $BA=E$. All my textbooks define the inverse $B$ of $A$ such that $AB=BA=E$. But I can't see why $AB=E$ ...
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65 views

If $A$, $B$ are $3\times 3$ matrices, and all elements are different from each other and greater in their absolute value than 3, then is $AB \ne 0$?

Let $A$ and $B$ be two $3\times 3$ matrices. All entries of $A$ are distinct and all entries of $B$ are distinct. All entries in both matrices are greater in their absolute value than 3. Then $AB ...
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109 views

Show that matrices are not similar

I have to show that the following matrices are not similar: $$A = \left[\begin{matrix} 1 & 3 & -3 \\ -3 & 7 & -3 \\ -6 & 6 & -2\end{matrix}\right]$$ and $$A' = ...
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What is the Best Way to See if Vectors are Equal?

Maybe this is a stupid question, but when I started to think about it I started to feel rather unsure. The question is what is the best way to see if vectors, or more specifically eigenvectors are ...
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230 views

Why can a matrix whose kth power is I be diagonalized?

Say A is an n by n matrix over the complex numbers so that A raised to the kth power is the identity I. How do we show A can be diagonalized? Also, if alpha is an element of a field of characteristic ...
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306 views

If $A^n=0$, then $I_n-A$ is invertible. [closed]

How do I solve this problem? $A$ is $n\times n$ and $A^n=0$. Prove that $I_n-A$ is invertible.
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any symmetric matrix is ​​invertible?

is a simply theoretical question, but any symmetric matrix is ​​invertible? i'm trying to prove this question but I don't know what I need to do. I apologize for the simple question but is a doubt ...
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455 views

How find the matrix $K$ such $AKB=C$

Question: Find a matrix $K$ such that $$AKB=C$$ given that $$A=\begin{bmatrix} 1&4\\ -2&3\\ 1&-2 \end{bmatrix},B=\begin{bmatrix} 2&0&0\\ 0&1&-1 \end{bmatrix} ...
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179 views

Fibonacci Calculation using a larger matrix

So the formula to generate the fibonacci sequence in matrix form is: $$ \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & ...
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4k views

On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?

I was seeing the proof that $\exp(A)\exp(B)=\exp(A+B)$ on link Show that $ e^{A+B}=e^A e^B$ where uses the hypothesis $AB=BA$? Thanks!
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finding determinant as an function in given matrix

Calculate the determinant of the following matrix as an explicit function of $x$. (It is a polynomial in $x$. You are asked to find all the coefficients.) \begin{bmatrix}1 & x & x^{2} & ...
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Why is $\frac{x}{\| x \|}$ a unit vector? [duplicate]

Let $x$ be a vector in $\mathbb{R}^n$. Why is $\frac{x}{\| x \|}$ a unit vector, for $x\neq 0$? If I try to simplify it, I get the following: $\frac{x}{\sqrt{x \cdot x}}$, and I'm not sure how to ...
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112 views

Is $\operatorname{GL}(n, \mathbb{R})$ with multiplication a group?

I am looking at an exercise that saying $GL(n,\mathbb{R})$ with multiplication, in other words the nxn matrices with real entries together with multiplication is a group. I wonder the following: do ...
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244 views

Proving that every $2\times 2$ matrix $A$ with $A^2 = -I$ is similar to a given matrix

Show that every $2\times 2$ matrix $A$, for which $A^2=-I$, is similar to $\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$. I need help proving this. Don't have any idea how to proceed. ...
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74 views

Given $C$ is a $2\times 2$ matrix and $C^3=0$, determine whether $C^2=0$.

Given $C$ is a $2\times 2$ matrix and $C^3=0$, determine whether $C^2$ is also $0$. Thanks for your help.
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1k views

How to prove that det($A^{T}A$) is nonnegative?

Why is the determinant of the product of a matrix and its transpose nonnegative?
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6k views

Intersection between two lines

I'm having a lot of trouble trying to figure this out. Our prof hasn't covered this and we our assignment is due in two days. I'm trying to figure this out on my own: Consider the two lines ...
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220 views

Matrix with $1$s in the diagonal and off diagonal entries with absolute value less than $1$ invertible?

I ran into this problem recently. If $A$ is a $n\times n$ matrix with $1$s in the diagonal and all off diagonal entries have absolute value less than $1$, is $A$ invertible? It it definitely true ...