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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Isomorphism matrix problem

So the question asks: Recall that $U^{2\times 2}$ is the vector space of 2X2 upper triangular matrices. Which of the following functions are isomorphisms? A. The function T: $U^{2\times 2}$ to ...
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25 views

How polynomials are represented in matrix form for Univariate Polynomial. [on hold]

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
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173 views

Invertible skew-symmetric matrix

I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the ...
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1answer
15 views

Uniqueness of the reduced echelon form - a doubt regarding the proof

Let's take a look at this proof It is claimed: It follows that both the $n$-th columns of $B$ and $C$ must contain leading 1's, for otherwise those columns would be free columns and we could ...
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135 views

Prove image of basis is basis of vector space

With $V$ and $W$ being vector spaces, and $T: V \rightarrow W$ being a linear transformation: c) Suppose $B: (v_1, v_2, \cdots, v_n)$ is a basis for $V$ and $T$ is one-to-one and onto. Prove that ...
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28 views

Explanation on derivation of this equation?

I'm extremely stuck on how my book was able to derive this equation. Basically, it says: Let $V = W = P_2(\mathbb{R})$. A basis for V is $1, 1+x, 1+x+x^2$. Define the linear transformation $T$ such ...
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Linear Algebra - Changing bases?

Let $T: \mathbb{R}^3\rightarrow M_{2\times2}(\mathbb{R})$ be the linear transformation defined by $T((a,b,c)) = \begin{bmatrix}a&5a\\c&3c\end{bmatrix}$. Consider the bases ...
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1answer
90 views

Which of the following are subspace in $\mathbb{R}^4$?

Which of the following are subspaces in $\Bbb R^4$? (a) $\mathrm{span}([1,2,-1-2],[0,2,1,0],[3,1,1,2],[1,2,1,3])$ (b) $\mathrm{span}([1,2,1,2,1],[2,1,2,3,5],[-1,1,0,0,1],[2,1,3,4,2])$ (c) The hyper ...
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82 views

Linear Dependence problem

Lets suppose we have a set of vectors $\{(1 ,0, 0, 0 ) , (0, 1, 0, 0 ), (2, 0, 0, 0 )\}$. By definition this set is linearly dependent because we can find constants $c_1,c_2, c_3$ ( such that all ...
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76 views

Row Reduce Augmented Matrix

I am having issues actually row reducing it. What I initially get for the augmented matrix is: \begin{pmatrix}\begin{array}{cccc|c} 0 & 1 & -2 & 1 & 2\\ 2 & -2 & 4 ...
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1answer
23 views

Eigenvalues of matrix summation

Let $A$ be symmetric positive definite matrix with eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$. Can we express the eigenvalues of $I-A$ using eigenvalues of $A$? I can't find properties of ...
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$A$ is a linear transformation with eigenvalues

Let $A:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be a linear transformation with eigenvalues $\frac{2}{3}$ amd $\frac{9}{5}$. Then, there exists a non zero vector $v\in\mathbb{R}^2$ such that ...
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1answer
33 views

Why are these functions linearly dependent?

$f1(x) = x$ $f2(x) = x^2$ $f3(x) = 5x - 4x^2$ From my understanding a set of functions are only linearly dependent if you can show that one function is simply a scaled version of another in the ...
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2answers
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Condition on matrix to ensure nontrivial Jordan canonical form

In my understanding, in order to ensure that a matrix $A$ has a nontrivial Jordan canonical form, one needs to come up with such a matrix whose geometric multiplicity is less than algebraic ...
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19 views

Linear Algebra Solving for Matrices

Consider the following matrix. $A$ = [ a b c ] [ d e f ] [ g h i ] Suppose that $\det(A) = −2$. Let $B$ be another $3 \times 3$ matrix (not ...
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20 views

Proving Linear Independence Given Odd Absolute Values

With three vectors $a,b,c \in \mathbb{R}^3$, the magnitude of a$,b,c,a-b,b-c$, and $c-a$ are all odd integers (not necessarily distinct). How could you prove the three vectors are linearly ...
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23 views

Basis of a Subspace given an Equation.

Hi, I was wondering if this question is asking us to find the basis of the kernel of this transformation from $\mathbb{R}^4 \rightarrow \mathbb{R}$ ? Thanks
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Is this how to do matrix representation?

Say, $$f: \mathbb{Q}[t]_{4} \to \mathbb{Q}[t]_{4}$$ $f(q)=3q'''+2q''$ And we have the base $B=\{1,t,t^2,t^3,t^4\}$ and we wanted to find $[f]_{B}^{B}$ Then is this what would we do; $$f(1)=0$$ ...
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74 views

What is the enclosed volume of an irregular cube given the x,y,z coordinates of the 8 corners?

I have the xyz coordinates of 8 points that forms an irregular-shaped cube. This is an animation, so the cube is undergoing periodic or cyclical shape-change. The co-planarism of any group or set ...
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1answer
24 views

Distance between projections

Let $x,y,z \in \mathbb R^2$ such that $||x|| = ||y||= ||z|| = 1$. Project $z$ onto the lines defined by $x$ and $y$ as follows: \begin{equation} z_x = (z^\text{T}x) x, \ \ z_y = (z^\text{T}y) y, ...
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1answer
21 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
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3answers
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Compute a natural number $n\geq 2$ s.t. $p\mid n \Longrightarrow p^2\nmid n$ AND $p-1\mid n \Longleftrightarrow p\mid n$ for all prime divisor p of n.

Question: Compute a natural number $n\geq 2$ that satisfies: For each prime divisor $p$ of $n$, $p^2$ does not divide $n$. For each prime number $p$, $p-1$ divides $n$ if and only if $p$ divides ...
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$\bf{x'}=Ax$ with eigenvalues of multiplicity greater than $1$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, if I solve it by first finding matrix $\bf{P}$ and then ...
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1answer
23 views

A linearly independent set that spans a space

So, in partial differential equations, we generate solutions for PDEs (kind of obviously). However, while the solutions we generate span the space of all solutions and are all linearly independent, ...
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37 views

Is it possible to find determinant of a matrix by given the eigenvectors and the eigenvalues

If I already found the eigenvalues and eigenvectors of a particular matrix , is there an easy way to find the determinant of that matrix ?
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25 views

How can I rearrange this formula to give it in terms of $t$? [on hold]

How can I rearrange the equation $$ e^{2t} = \frac{y^{2}(y+1)}{y-1} $$ to give it in the form $y = f(t)$?
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36 views

Linear Algebra. Is this question realte to combination and factorials?

I am not able to understand this question and what is the entries of matrix A exactly. Question Thanks.
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1answer
10 views

Subspace of $C^3$ that spanned by a set over C and over R

Given $A=$ $\left\{ {(1,2 + i,i),(1,3 + i,3 - i),(i,3i,4 + i)} \right\}$ Let $SP_CA$ be the linear space spanned by A over $C$ Let $SP_RA$ be the linear space spanned by A over $R$ what is the ...
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What is $\left | \left | A \right | \right |$ equals to in linear algebra?

Can someone please tell me what is this $\left | \left | A \right | \right |$ equals to? (determinant inside determinant)
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Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
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21 views

Show linearity of this map, if and only if statement

Let $Γ_f = \{(s, t)\mid t = f(s)\} \subset S \times T$ Suppose that $U$ and $V$ are vector spaces and $f ∈ V^U$ (i.e. f is a map of underlying sets which is not necessarily linear). how do you show ...
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1answer
27 views

Linear Algebra - properties of positive semidefinite matrix

Let $A=(a_{ij})$ be a positive semi-definite, symmetric matrix, of order $3\times 3$ satisfying: $$ \Sigma_{j=1}^{3} a_{ij}=0 $$ for $i=1,2,3$ (i.e.- the sum of each row is zero). Prove: ...
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1answer
34 views

Linear algebra: Solving a system of equation matrix with a variable as coefficient.

Let's consider this augmented matrix $$\left(\begin{array}{ccc|c} 3 &-6 &6 &15\\ -2 &7 &a &-25\\ 2 &-6 &6 & 20 \end{array}\right)$$ I'm trying to ...
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1answer
551 views

Covariance- v. correlation-matrix based PCA

In principal component analysis (PCA), one can choose either the covariance matrix or the correlation matrix to find the components. These give different results because, I suspect, the eigenvectors ...
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Bounding the perturbation between eigenvectors

Can somebody explain this part of the proof of a deduction from the Davis-Kahan $\sin \theta$ theorem? I understand how to get from: $||P_{u_1} - P_{v_1}|| \le \epsilon$ to $||P_{u_1}v_1 - v_1|| \le ...
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Bounding entries of random vector

Given a random vector $\mathbf{e} \in \mathbb{R}^n$, is it possible to count (or bound) the number of entries in $\mathbf{e}$ that have $|e_i| \ge 1/ \sqrt{n}$? It is known that entries in ...
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2answers
80 views

What really is diagonalization?

If I have a square matrix $A$ representing a linear transformation $T:V\rightarrow V$ w.r.t the basis, $B=\{v_1,v_2..,v_n\}$ and $A$ is Hermitian. So we have $Av_n=\lambda_{n}v_n$ where ...
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1answer
43 views

Linear Alg. Short proof on determinant

Hi can I get a quick check on my proof to see if it is correct. proof
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1answer
18 views

Identity regarding partial derivatives and polar representation

Let $f(x,y)$ be a differentiable function, and $g(r, \theta) = f(r \cos \theta , r \sin \theta)$. I need help showing that: $$ \left( \frac{ \partial f}{\partial x} \right)^2 + \left( \frac{ \partial ...
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3answers
52 views

Linear vs. bilinear

I'm tripping over something elementary: Suppose $f:\mathbb{R^2}\rightarrow X$ is linear, then $f(x+y)=f(x)+f(y)$ for all vectors $x$ and $y$. Now suppose that $f$ is also bilinear and in particular ...
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$f: L \to L$ a diagonalizable operator with simple spectrum and $g: L \to L$ s.t $gf = fg$

I am making the exercises of Kostrikin and Manin (Linear Algebra and Geometry) and it has this question that I can't solve. Let $f: L \to L$ a diagonalizable operator with simple spectrum and $g: L ...
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Distance between two symmetric equations

I have been requested to solve this problem: Compute the distance between the lines: $L_{1}:\frac{x-2}{3}=\frac{y-5}{2}=\frac{z-1}{-1}$ and $L_{2}:\frac{x-4}{-4}=\frac{y-5}{4}=\frac{z+2}{1}$ This ...
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2answers
41 views

Invertible matrices modulo $29$. [duplicate]

Consider the $n\times n$ matrices with elements in $\mathbb{Z}_{29}$. How many of these are invertible? In total there are $29^{n^2}$ matrices of of dimension $n\times n$. Now I need to find how ...
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2answers
44 views

Expressing linear transforms using linear functionals: is this possible?

Work over a fixed but arbitrary field. Let $Y$ and $X$ denote finite-dimensional vectorspaces, and let $y \in Y^n$ denote a sequence of elements of $Y$, where $n$ is a natural number. It seems ...
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1answer
58 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)? Sorry my calculus is a little rusty. I ...
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“linear independence” of characters? Artin's theorem.

Let $G$ be a monoid and $K$ be a field. We define a character of $G$ in $K$ as a monoid homomorphism $f\colon G\to K^{\times}$, where $K^{\times}$ denotes the multiplicative group of $K$. One fact ...
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Let $\operatorname{rank} A=1$ then there are, $x,y\in \mathbb{C}^n$ such that $A=xy^T$ [on hold]

Let $A\in M_n$ and $\operatorname{rank} A=1$. Are there $x,y\in \mathbb{C}^n$ such that $A=xy^T$?
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871 views

Proving the dimension of the intersection of 2 subspaces

Assume that $U$ and $W$ are distinct subspaces $( U ≠ W )$ of a four-dimensional vector space $V$ and $\dim(U) = \dim(W) = 3$. Prove that $\dim ( U ∩ W ) = 2$.
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Find the cardinality of a subset of $GL_n( \bf F_p)$

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$ X_m = \{A \in GL_n( \bf F_p): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear ...
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Why is Gaussian matrix full rank?

Suppose $A\in R^{n\times n}$ is a matrix with independent standard normal entries. Is there an elementary argument to show that $A$ is nonsingular with probability $1$?