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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Skew-symmetric matrix subspace dimension and basis

If $M$ is the vector space of $2\times 2$ real matrices, then I can show that $$ \{A \in M \mid A^\mathrm{T}=-A \} $$ is a subspace of $M$, since $$ \left[ \begin{array}{cc} x & z \\ -z & ...
0
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1answer
46 views

Finding the dimension of a vector space

I have done part i by putting $v_1,v_2,v_3,v_4$ into a matrix and the putting that matrix intro REF to find value of constants $c_1$ to $c_4$ But for the next part how do i find the dimension of $W$ ...
0
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1answer
19 views

Finding determinants with properties given minimal information

Let $A$ and $B$ be $2 \times 2$ matrices, where $\det(A)=2, \det(B)=3$. Find $$\det(B^{T} A^{-3}(2BA)^2)$$ I know that $\det(B)$ and $\det(B^{T})$ are the same so it stays $3$. I don't quite know ...
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1answer
72 views

Find the Jordan form of a 4 x 4 matrix

Find the Jordan Form of $$ A=\left[\begin{array}{cccc} 0 & -16 & 0 & 0\\ 1 & 8 & 0 & 0\\ 0 & 0 & 0 & -6\\ 0 & 0 & 1 &5 \end{array}\right] $$ First, ...
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2answers
36 views

finding determinants using different properties

The equation is as follows: $\operatorname{det}(2A^{-1} + 7\operatorname{adj}(A))$ Here I know that $\operatorname{det}(A^{-1}) = (\operatorname{det}(A))^{-1}$ and $\operatorname{det}(kA) = k^n \...
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2answers
32 views

finding determinants using properties

I have a problem here dealing with matrix determinants. Let $A$ and $B$ be $2\times2$ matrices, where $\det(A) = 2, \det(B) = 3$. Find: $\det((2A)^{-1} B^2)$ So far I have a formula that says ...
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2answers
53 views

Orthogonal Projection in subspace

Consider the vector space $\mathbb{R}^n$ with usual inner product $<.,.>$. Take $Y\in \mathbb{R}^n$ and $X \in \mathbb{R}^n$ such that $Y=[y_1,y_2,..y_n]^t$ and $X=[1,1,....1]^t$ ...
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1answer
52 views

prove by mathematical induction involving a root [duplicate]

I am finding this problem rather intriging on the k+1 part of it... Prove by induction that...
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1answer
296 views

Find the coordinate matrix of a polynomial with respect to a non-standard basis

I'm stuck on this question here: Find the coordinate matrix of $2-4x-3x^2$ with respect to $B = {2, x^2-1, 1-2x-x^2}$ I did the following: $a(2) + b(x^2 - 1) + c(1-2x-x^2) = 2-4x-3x^2$ But now I'm ...
2
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2answers
107 views

What is the difference between $n$-tuples, $m \times 1$ and $1 \times n$ matrices?

Isn't the tuple different structure from $m \times 1$ or $1 \times n$ matrix? Why can you mix them?
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1answer
72 views

Steinitz exchange lemma - basis

I've read on the internet this definition of Steinitz exchange lemma: Steinitz exchange lemma Let $S = \{v_1, \ldots , v_m\}$ satisfy $\mathrm{Span}(S) = V$ and let $T = \{w_1, \ldots , w_k\}...
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1answer
41 views

Conditional expectation, pinching

Let $\mathfrak{C}$ be a unital $*$-subalgebra of the full matrix algebra $M_n(\mathbb{C}).$ Let $\mathbb{E}_\mathfrak{C}$ be the orthogonal projection from $M_n(\mathbb{C}),$ endowed with the ...
0
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2answers
47 views

Solving sum of poisson

I'm trying to solve this summation but I got stuck at the last step. Hope anyone could help me with this algebra. $\sum\limits_{n=1}^\infty 10000\cdot(n-1)\cdot\frac{1.5^n\cdot e^{-1.5}}{n!}$
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1answer
38 views

The significance of a basis of a vector space.

I find myself quite confused about bases of vector spaces, and exactly what they represent. For if you consider the $3D$ real space, it is $3-$tuple of real numbers. In a geometrical context, what is ...
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1answer
86 views

Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$.

Let , $A_{6\times 6}$ diagonal matrix with characteristic polynomial $x(x+1)^2(x-1)^3$. Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$. From characteristic polynomial of $A$ , first we ...
2
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1answer
27 views

On the subset of a closed vector subspace

Theorem: Let $H$ be a Hilbert space, and let $U$ and $V$ be closed subspaces of $H$ such that $U\subset V$. Then there exists a nonzero vector $v\in V\backslash U$ such that $v\bot U$. The fact that ...
0
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2answers
84 views

if $rank{(A - \lambda I)^k} = rank{(B - \lambda I)^k}$ then $A$ is similar $B$

Let $A,B \in M_n(\mathbb{R}).$ Suppose for all $\lambda \in \sigma (A)$ and for all $k \geq 0,$ we have $\mathrm{rank}(A - \lambda I)^k = \mathrm{rank}(B - \lambda I)^k.$ Then why are $A$ and $B $ ...
0
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1answer
62 views

If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ - 1}}$

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
2
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1answer
59 views

A set is linearly dependent if and only if there is a proper subset with the same span?

Let $S$ be a subset of a vector space. I conjecture that $S$ is linearly dependent if and only if there exists a proper subset $S' \subset S$ such that $\operatorname{span}(S')=\operatorname{span}(S)$....
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1answer
61 views

Jordan form of different matrices

Suppose you have a 4x4 matrix with the characteristic polynomial equal to the minimal polynomial $m_F(x)=C_F(x)=(x-3)^2(x+2)^2$. Find the Jordan form. Is this the correct solution? $$ M=\left[\begin{...
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1answer
68 views

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces Ok so I know I have to use Baire's Cathegory Theorem here. And I've done the following, lets suppose on the ...
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1answer
134 views

Limmiting solution of $Ax=b$ to positive quantities

My personal trainer put me on a diet recently which has had me tracking the macro-nutrients that I eat i.e. protein, carbohydrates and fat. I am supposed to eat a specific amount each meal and eat ...
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1answer
38 views

understand an answer to linear span of polynomal subspace exercise

i am looking at an answer to an exercise who asks to find a linear span for, and I don't really understand the solution $$p(x) = ax^3 + bx ^2 + cx + d$$ and this is the solution i see $$ p(x) \in M ...
25
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0answers
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Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
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1answer
79 views

find equality between linear spans

$$U = Sp\{(2,5,-4,-10), (1,1,1,1),(1,0,3,5), (0,2,-4,-8)\}$$ $$ W = Sp\{(1,-2,7,13), (3,1,7,11), (2,1,4,6) \}$$ two questions: prove that $U = W$ find the values of the $a \in \mathbb{R}$ where the ...
15
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4answers
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Vector Spaces: Redundant Axiom?

Question Why are the axioms for vector space independent? More precisely $1x=x$ seems redundant... (I take the axioms from: Wikipedia) Explanation One has for zero vector: $$\lambda0+\lambda0=\...
0
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3answers
97 views

What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots…

I found this question in a maths-group in Facebook- What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots........ I think we do not count repeated roots ...
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2answers
31 views

linear span of subspace

we have the following subspace over $\mathbb{R}$ $$M = \{ A \in M^{{n\times n}} | A = -\overline{A} \}$$ I found that it is a subspae and now I need to find the linear span of it. how can I calculate ...
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2answers
141 views

What are the “building blocks” of a vector?

Lets say I have a set of vectors $V$ that includes this vector: $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ I interpret it as $x = 1, y = 2, z = -1$ (that being three dimensions for this vector). I know ...
3
votes
2answers
150 views

Is there always a bijection between the subspaces of dimension m and codimension m in a finite dimensional vector space?

Say $U$ is a vector space of dimension $n$ over $\mathbb{F}$, if $\mathbb{F}$ is finite we know there exists a bijection between the subspaces of dimension $n-m$ and $m$. Can this be generalized to ...
0
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1answer
33 views

Commuting polynomials of a matrix

I am proving a proposition and I notice that I can do it easily if this proposition is true: Let $f(t), g(t)$ two polynomials and $A\in M_n(\mathbb {F})$ a matrix such that $f(A)$ is non-singular. ...
0
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1answer
48 views

Representation of the inverse of an variance-covariance matrix $\hat{\Sigma}^{-1}$

Given $T$ observed vectors $x_i\in\mathbb{R}^N, i\in\{1,\ldots,T\}$. Define $\hat{\Sigma}$ as the corresponding empirical covariance-matrix of the Observations $X=\left(\begin{array}{c} x_1' \\ \...
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2answers
31 views

Necessary and sufficient condition for a bilinear form to be symmetric

Given the bilinear form $f(A,B)=\operatorname{tr} (A^t M B)$ where $A,B$ are two $n\times n$ matrices I have to find a necessary and sufficient condition (on $M$) for $f$ to be symmetric. I found out ...
2
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0answers
86 views

Quantum Teleportation - how to prove the general case?

I've taken a course of quantum information theory and although I can compute a quantum teleportation in an explicit case where I'm given a quantum entanglement shared by Alice and Bob (normally 1/root(...
5
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2answers
51 views

Simple question about quadratic form

Suppose $V$ is a vector space over $\mathbb{C}$ and $\dim V \geq 2$. I must prove that if $q: V \to \mathbb{C}$ is a quadratic form then there exists $v\neq 0$ such that $q(v)=0$. Also, how the answer ...
6
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1answer
127 views

What does this vector notation really mean?

With regard to vectors, how is this (form 1): $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ Different to this (form 2): $$\begin{bmatrix}1\ 2\ -1 \end{bmatrix}$$ I would think that the first set consists ...
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1answer
48 views

How can we find if a matrix is full column rank

If $A$ is an $n*k$ matrix with complicated form of elements. How can I show this matrix is full column rank? By complicated form I mean there is no known form for the elements of $A$.
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2answers
104 views

Find Jordan Form of αA (α is a scalar, A a matrix)

In my linear algebra course I have a problem which goes as follows: Suppose A is an nxn matrix over field (R) And J(A) is the jordan form of A. Given α belongs to field R, what is the jordan form of ...
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1answer
62 views

Why can't I get this with Gram-Schmidt

Can someone assist me with a very simple problem. I cannot get these two vectors to be orthogonal using Gram-Schmidt: $\{(1,-1,1),(2,1,1)\}$ What am I doing wrong? Let $v_1=(1,-1,1)$. $...
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1answer
20 views

Simplifying an unusual quadratic linear algebra expression

I came across the following expression when solving a maximisation problem. I have the following ingredients: Matrices $\Omega, P \in \mathbb{R}^{n \times n}$ Vector $t \in \mathbb{R}^n$ Also let $\...
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1answer
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Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)? .

Let $A \in {M_n}$.Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)?
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0answers
56 views

question about isotropic subspaces

if $V$ is a complex vector space of dimension $2n$ and $Q$ a bilinear form over $V$, the definition of an isotropic subspace is the following: $$\Lambda:Q(\Lambda,\Lambda) \equiv 0$$. Suppose that $\...
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1answer
34 views

Proving vectors as a basis in $E^{m}$

Show that if the vectors $a_{1}$, $a_2$, $\cdots$, $a_m$, are a basis in $E^{m}$, the vectors $a_{1}$, $a_2$, $\cdots$, $a_{p-1}$, $a_{q}, a_{p+1}, \cdots,a_{m}$, also are a basis if and only if $y_{p,...
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0answers
149 views

Linear Algebra- Subspace Question

"Is the set of all polynomials over the real numbers of degree exactly 2 a subspace of P∞(R)?" So apparently the answer to this is 'no'. Can anyone explain why?
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2answers
63 views

Finitely generated modules, related to the Gaussian integers

Let $M = \mathbb{Z}[i] = \{a + bi|a, b \in \mathbb{Z}\}$ be the additive group of the Gaussian integers. Consider the $\mathbb{Z}$-submodules $N_1 = 2M$, $N_2 = (1 + i)M$. I now want to figure out ...
2
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1answer
50 views

How do I derive this polygonal function from sample values?

I have 4 parameters with 16 sample data points each. When I plot them, I get this: The curves lead me to suspect that all these of 64 data point are derived from one polygonal function with 4 ...
3
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2answers
83 views

Proving the distance between two points is 1 on a continuous function

this is a tough question. We suppose $f$ is continuous on $[0,2]$ and $f(0) = f(2)$. We want to prove $\exists$ $x, y \in [0,2]$ such that $ \lvert y-x \rvert = 1 $ and $ f(x) = f(y)$. My attempt ...