Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Invertible Matrix and Linearly Independent Vectors Proof

Trying to do this one: Suppose $A$ is an invertible $n$ x $n$ matrix and the vectors $v_1$, $v_2$, ..., $v_n$ are linearly independent. Show that the vectors $Av_1$, $Av_2$, ..., $Av_n$ are linearly ...
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How can I show that an odd degreed polynomial with coefficients in the real space always has a root in $\mathbb{R}$? [closed]

How can I show that every odd degreed polynomial with coefficients in the real space will have a real root?
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90 views

Find $\frac{a^3}{a^6 + 1}$ given a is a root of a quadratic equation

My question is: If a is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$. So, I figured we can use the Sridharacharya ...
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5answers
149 views

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable?

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable? I didn't succeed to get any information about it. Could anyone explain please?
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73 views

What’s the sum of four vectors that form a loop?

4 vectors that connectod end to end. what is the sum of all vectors ?
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Find the determinant, assuming that

Given that $$\begin{vmatrix} a & b & c \\ d & f & g\\ q & w & e \end{vmatrix} =5$$ It is a whole matrix above. $$\begin{vmatrix} a & b & c \\ d & f & ...
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1answer
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Is this still considered $\{0\}$

Is the following solution to the matrix a zero subspace? (Assume that the last column of zeros is the constant portion of the matrix) I'm working on some kernel problems, and if a linear ...
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598 views

Let a, b, c be three nonzero vectors, any two of which are perpendicular. Prove that these 3 vectors are linearly independent.

Here is my answer: We assume two cases, and work to prove that both assumptions are incorrect, leading to a proof. Without loss of generality, assume that $ \mathbf{a} $ is linearly dependent with ...
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$T\colon \mathbb R^2 \rightarrow \mathbb R^2$ be a linear transformation with $T(x)=0$ for all $x$ such that $||x||=1.$

I came across the following problem that says: Let $T\colon \mathbb R^2 \rightarrow \mathbb R^2$ be a linear transformation. Assume that $T(x)=0$ for all $x$ such that $||x||=1.$ Then, which ...
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997 views

$V$ is isomorphic to $V^{\ast\ast}$, the double dual space of $V$.

Prove that for any vector space $V$ the map sending $v$ in $V$ to (evaluation at $v$) $E_v$ in $V^{**}$ such that $E_v(\phi) = \phi(v)$ for $\phi$ in $V^*$ , is injective. Derive from this that if ...
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194 views

The relation between the rank of $AB$ and $A,B$?

Assume, we know that $A,B\in M_{n}(\mathbb R)$. Can we say that: $$\mathrm{rank}(AB-BA)\leq [n/2]$$ Thanks for any hint.
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Is the map linear?

$$\begin{bmatrix}a&b\\\\c&d\end{bmatrix}\mapsto \begin{bmatrix}ad-bc\\\\0\\\\0\end{bmatrix}$$ If it is linear I need to find a basis for the kernel and image but I am struggling to do this so ...
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627 views

Linear algebra proof - invertible matrices

I need to set up a proof for this problem: Given that $A$ and $B$ are both $n\times n$ matrices. $A$ is invertible, and $AB=BA$. Prove that $A^{-1}B=BA^{-1}$. I'm just unsure how ...
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1k views

Relationship between column space and image (linear algebra)

I'm having trouble understanding this. Why does the column space of the matrix of a linear transformation equal the image of the linear transformation? I know the answer is really simple...but it ...
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595 views

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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365 views

Intersection of subspaces

I am working on example of vector space. I have question: Let $\{V_1,V_2,\ldots,V_t\}$ be a family of $n$-dimensional subspaces and the dimension intersection of any $n$ subspace is at least one. Is ...
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169 views

is $U^{2}=P$? where $U$ unitary and $P$ orthogonal projection

Problem: I am trying to solve the following problem, but I couldn't. The problem is: Let $U$ be unitary matrix. Let $P$ and $UP$ be orthogonal projections. Is it true that $U^{2}=P$? If yes, please ...
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How is it shown that a Hermitian matrix will be positive definite?

The following Hermitian matrix is given for all $m \in \mathbb{Z}$: $$A_m = \left( \begin{array}{cc} m & i \\ -i & m\\ \end{array} \right)$$ (i) Show that $A_1$ is not ...
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How to prove that convolution is associative?

How to prove that convolution is associative, that is $(x*y)*z = x*(y*z)$, for $x,y,z \in l^2(Z_N)$. Either directly from the definition of convolution( Definition of convolution: For $z,w \in ...
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6answers
301 views

How to solve a system of two linear equations with two unknowns?

How do I solve this system of equations? $$\begin{cases} 7(a+b)=b-a \\4(3a+2b)=b-8\end{cases}$$ Progress I tried both substitution and elimination, but when I set $a$ or $b$ free on one side, I ...
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$f( \alpha x + \beta y) =\alpha f(x) + \beta f(y)$

My friend had the first Math 101 Class in University, and he really finds its hard and he says the lecturer didn't teach this well. Can anyone give me a proof for this property in linear algebra- $f( ...
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$A^{T}A$ is diagonal. What can I say about $A$?

Is there any special property about the elements of $A$ if $A^{T}A$ is diagonal? I imagine you need some sort of symmetry but I can't see what it should be. Edit: Sorry, maybe it's better phrased ...
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Example: Algebraic Multiplicity vs Geometric Multiplicity

Is there a simple example of a matrix having an eigenvalue whose geometric multiplicity is strictly smaller than its algebraic multiplicity?
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114 views

How to show that the set of functions $1,x,x^2,x^3…$ is linearly independent? [closed]

Show that the set of functions $1,x,x^2,x^3...x^n...$ is linearly independent on any interval $[a,b]$. If $$c_1+xc_2+x^2c_3+x^3c_4...=0$$ we should show $$c_i=0,\quad i=1,2, \ldots$$ how could I ...
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for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
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4answers
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Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
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4answers
70 views

Compute $\det(tI - M)$

I've been struggling computing $\det|tI - M|$ . Given $M=\ \begin{bmatrix} 3& 2& 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \\ \end{bmatrix} $ It has ...
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4answers
86 views

What am I doing wrong here?

Consider this system of equations: $$ \begin{cases} x+y=6\\x-y=5\\2x+3y=7 \end{cases} $$ This is an overdetermined system and doesn't have a solution (the 3 lines don't intersect). But by adding 2nd ...
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69 views

Why is the Det(a)=0 not a subspace? [closed]

I'm reading my linear algebra textbook, and it says word for word: The following sets is not a subspace when the set of all 2x2 matrices B such that det(B)=0. I just need help trying to understand ...
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For which values of $k$, we have $A = A^{-1}$?

I got this question in hw. Can anyone help me solve it? Let $ A = \left( \begin{array}{ccc} k & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & k \end{array} \right) $ For which values of ...
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Existence of a unique polynomial

Suppose $z_1, .....z_{m+1}$ are distinct elements of $F$ and $w_1,....,w_{m+1} \in F$. Prove that there exists a unique polynomial $p \in P_m(F)$ such that $p(z_j) = w_j$ for j=1,...m+1. Any ideas on ...
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62 views

About matrix products $A^{T}A$ and $ AA^{T} $

I'm investigating the relationship between 2-norms and eigenvalues of $A^{T}A$ and $ AA^{T} $, in order to better understand the SVD decomposition. How can I prove that $A^{T}A$ and $ AA^{T} $ are ...
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113 views

If $AB=BA$ then they are diagonal

I saw this statement If $AB=BA$ then both matrices must be diagonal. Why is that?
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74 views

How to find $\det(-6A)$, if $\det A=-4$?

How do I solve this? Assume that $A$ and $B$ are $6 \times 6$ matrices, such that $\det A = -4$ and $\det B = -2$. Find $\det(-6A)$.
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How find this $\det(M_{n})$

Let $n$ be a postive integer, let $a,b,c$ be real numbers, with $a\neq b$, and let $M_{n}$ denote the $2n\times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by $$m_{ij}=\begin{cases} x & ...
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60 views

If $\operatorname{rank}(AB)=n$, what are the $\operatorname{rank}(A)$ and $\operatorname{rank(B)}$?

$A$ and $B$ are $n\times n$ matrices. Any hints on how to solve this or where to find the answer are welcome
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Prove the following

Prove that if $p$ and $q$ are polynomials over the field $F$, then the degree of their sum is less than or equal to whichever polynomial's degree is larger $$\deg(p+q)\leq \max \left\{\deg(p),\deg(q) ...
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1answer
410 views

What does this linear algebra notation mean?

I'm trying to prove that a particular $V$ is a $\Bbb{Q}$-vector space. The question says to take the element $0_V = 1$, the function $+_V : V \times V \to V$ given by the function $[x +_V y = xy]$, ...
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139 views

Finding determinant for a matrix with one value on the diagonal and another everywhere else

Let us look the the matrix $\left(\begin{array}{ccccc} a & b & b & b & b\\ b & a & b & b & b\\ b & b & a & b & b\\ b & b & b & a & b\\ ...
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2answers
49 views

Prove the existence of a multiplicitive Inverse

Let $F$ be a field, such that $$F=\{a+b \sqrt{2}\}$$ Such that a and b are rational numbers. Prove there exists a multiplicative identity. I just expanded the product of two elements, and collected ...
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62 views

Why is this vector-set linear-dependent?

$$\{ (1,0),(0,1),(0,0)\} $$ Maybe I'm losing it, but I can't see here a vector which is a linear-combination of the other two.
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If $\textrm{dim}(V)<\infty\Rightarrow \textrm{dim}(U)<\infty$?

Consider $U$ and $V$ two vector spaces over a given field. $T:U\longrightarrow V$ and $S:V\longrightarrow U$ two linear operators such that $S\circ T=Id_U$. If $\textrm{dim}(V)<\infty$ how can I ...
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Is there any formula for summation?

$$0.01\sum_{x=1}^{30}(0.99)^{x-1} = 1-0.99^{30}$$ I wonder if there is a formula for summation and I want to know. Would anyone mind telling me? It would be better for me to solve problems, like the ...
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48 views

eigenvalues of $AB$ are eigenvalues of $\sqrt{B} A \sqrt{B}$

Suppose $A,B$ are symmetric positive definite matrices. An author claims that the spectrum of $AB$ is the spectrum of $\sqrt{B}A\sqrt{B}$. Why? Certainly they have the same trace by cyclic ...
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191 views

Understanding Span, Basis, and Dimension

I am a bit confused with span, basis, and dimension (when dealing with vector spaces). My teacher told us that a span is a finite linear combination. And I know that a basis is a spanning, linearly ...
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1answer
42 views

Find characteristic polynomial of $A+I$ if is knowing characteristic polynomial of $A$

Let $A \in \mathcal{M}_{3 \times 3}$ and let $x^3 - x$ be characteristic polynomial. Determine characteristic polynomial of $A+I$. We have eigenvalues of $A$: $0,1,-1$ so $A$ is similar to ...
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About $2\times 2$ similar matrices…

Let $A$ and $B$ be $2\times 2$ matrices with the same trace and the same determinant. Are $A$ similar to $B$? I know that they have the same characteristic polynomial. So, exist $P,Q\in GL_2(F)$ ...
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4k views

How to Tell If Matrices Are Linearly Independent

If I have two matrices, for example: $\begin{pmatrix}1&0\\2&1 \end{pmatrix}$ and $\begin{pmatrix} 1&2\\4&3\end{pmatrix},$ how do I determine if they are linearly independent or not in ...
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3answers
144 views

If a linear map $A$ is injective, then then there exists $c$ such that $|Ax|\geq c|x|\;\;\forall x$

If a linear map $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$ is injective, then there exists $c>0$ such that $|Ax|\geq c|x|$ for all $x\in\mathbb{R}^m$ Could someone give any solution or hint? ...
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110 views

$GL(n, \mathbb{C})$ is algebraically closed? [closed]

Let $GL(n,\mathbb{C})$ the group of non-singular matrices. Is it algebraically closed? For $GL(1,\mathbb{C})$ is it true; but if I take linear combinations of elements in $GL(n,\mathbb{C})$ with ...