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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2
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2answers
25 views

Find the slope of the line that goes through the given points

I know the formula for this type of problem is the second y coordinate subtracted from the first y coordinate over the second x coordinate subtracted from the first x coordinate but for the numbers ...
0
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2answers
32 views

Trouble finding Jordan Normal form for $4 \times $ 4 matrix

$M = \left(\begin{array}{cccc}0 & 1 & 0 & 0 \\-3 & 4 & 0 & 0 \\2 & -1 & 2 & 0 \\-1 & 1 & 1 & 2\end{array}\right)$. I find the eigenvalues to be ...
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1answer
24 views

Is Basis of a vector space a subset of the vector space

Now, I was going through my notes which says that basis of a vector space V is a set S such that 1)S is a linearly independent set 2)v=L(S) Now there might be multiple basis of a vector space.Hence ...
1
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1answer
22 views

Writing a matrix in terms of a basis

I've looked for examples but found none similar to this; I have $\mathfrak{sl}(2,K)$ with the given basis $S$ as follows: $S=\{e,h,f\}$ where $e = \pmatrix{0 & 1 \\ 0 & 0}$ $h = \pmatrix{1 ...
2
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0answers
19 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
1
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1answer
18 views

Linear Algebra Orthogonality Help

I am struggling with this one exercise from self-learning. I simply do not understand what it is asking. If someone could walk me through this problem I would be very grateful.
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0answers
20 views

Expanding linear functional to base of $V^*$

Given a linear functional $f_1\in V^*$ where $V^*$ is a dual space of $V$, I can expand it to the base of $V^*$ : $B^*=\{f_1,f_2,...,f_n\}$, that I know. But does it mean that exist a base $B$ for $V$ ...
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0answers
32 views

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. [on hold]

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. I understand that the solution to $\ker A$ is the set of all solutions to $Ax=0$. I'm confused about how I ...
-4
votes
1answer
28 views

Lift-club rates (This should be really easy)

Right, this is actually a real-life problem. I want to join Bob and Joe's lift club. Joe usually pays about \$40 a week (in total) to drive between A and B (for fuel). (Driving from A to B and back is ...
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0answers
13 views

Tensor algebra becomes a graded $R$-algebra short proof

I had a post proving that the tensor algebra becomes a graded ring, i have come up with a simple approach that goes as follows: Proposition: The tensor algebra $T(M)$ with multiplication defined ...
2
votes
1answer
32 views

Does a simplex with equal altitudes have to be equilateral?

Consider a simplex in $\mathbb{R}^d$. Assume that all its altitudes have the same length. Does it necessarily mean that the simplex is equilateral, i. e. all distances between its vertices are equal ...
2
votes
1answer
33 views

Least common multiple for integer matrices

Given two full-rank $3\times3$ integer matrices $M_1$ and $M_2$, I am trying to find integer matrices $N_1$ and $N_2$ such that $M_1N_1$=$M_2N_2$, such that $\left|\det(M_1N_1)\right|$ is minimal. ...
1
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2answers
22 views

Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that $T$ is bijection >iff T is injective.

Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that (a)$T$ is bijection iff (b)T is injective. Solution: show $(a)\implies(b)$ If $T$ ...
1
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1answer
32 views

I need help with this linear transformation.

Please let me know if my process or thinking is incorrect at any point. Let $T:P_3 \rightarrow P_3$ be the linear transformation such that $$T(-2 x^2)= 3 x^2 + 3 x,\\T(0.5 x + 4)= -2 x^2 - 2 x - 3, ...
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0answers
13 views

Write F+K as a span of some basis.

Let $F = \{(a,a+b,4b,0) | a,b\in \mathbb{R}\}$ and $K = \{ (c,2c+d,4c-d,2d) | c,d\in \mathbb{R}\}$. Write F+K as a span of some basis. Solution: The basis of $F$ is $\{(1,1,0,0),(0,1,4,0)\}$. ...
0
votes
1answer
56 views

Av = Bv all v implies A = B? [on hold]

If A and B are two $4\times3$ matrices such that $A\mathbb{v}=B\mathbb{v}$ for all $\mathbb{v}\in\mathbb{R}^3$, then matrices A and B must be equal. If it's not true, can you give me an example of ...
1
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1answer
32 views

Hyperplanes divide space

Problem. What is maximal number of connected components on which $n$ hyperplanes divides $\mathbb{R}^m$ if they all have 1 common point. In fact this problem was firstly stated in $\mathbb{R}^3$ and ...
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0answers
17 views

If $\dim(V) = 5$ and $\dim(W_1)=3$, and $\dim(W_2)=4$. $W_1\nsubseteq W_2$. Find $\dim(W_1+W_2)$ and $\dim(W_1\cap W_2)$.

Is my solution correct? Let $V$ be a vector space. Let $W_1$ and $W_2$ be subspace of $V$. Then $W_1+W_2 = \{a+b\mid a\in W_1 \text{ and } b\in W_2\}$. If $\dim(V) = 5$ and $\dim(W_1)=3$, and ...
1
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1answer
22 views

Prove that for every $k$ there's an invariant subspace

Let $V$, a vector space above $\mathbb{C}$ and let $T:V\to V$, a linear transformation. Show that for every $0\le k \le n$ there is an invariant subspace of $T$ with a dimension $k$. It seems ...
0
votes
1answer
24 views

matrix normal form

How can I prove that any $2\times2$ complex matrix is similar to a unique matrix of the form:$$\begin{bmatrix}0 & 1\\\theta_1 & \theta_2\end{bmatrix}$$ $\theta_i$ are complex. I tried to use ...
1
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1answer
34 views

Characterization of normed space

Let $\mathbb R^n$ be vector space over $\mathbb R$. Then we know that all norms over $\mathbb R^n$ are homeomorphism. Is it true for $\mathbb Q^2$ over $\mathbb Q$ For instance are the Euclidean norm ...
0
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1answer
22 views

How to show eigenvalues and singular values are the same?

Let $A$ be Hermitian and positive semi-definite i.e. $x^TAx \ge 0$ Let $K=A^TA=A^2$ $Ax=\lambda x$. Then $Kx=A^2x=\lambda^2 x$ Then, every eigenvector $x$ of $A$ is also eigenvector of $K$ with ...
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0answers
8 views

On non-degenerate bilinear forms on infinite dimensional vector spaces

For any non-degenerate bilinear form $(\cdot,\cdot)$ on a vector space $V$ and a linear functional $f$, there exists $v \in V$ such that $f(v)=(v,w)$ for all $w \in V$. It's easy in ...
0
votes
1answer
26 views

Changing the basis of a transformation

Given $T: \mathbb{R}^2 \to \mathbb{R}^2 : T\begin{bmatrix} x \\ y \end{bmatrix} \to \begin{bmatrix} 2x+y \\ x-3y \end{bmatrix}$ with standard basis $\mathcal{B}$ and basis ...
1
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1answer
31 views

$V=ker\ (T)\oplus im\ (T)$ if $T$ is a self-adjoint

Let $V \;$ be an inner product space over a field $\Bbb{F}$ and let $T:V\to V$ be a self-adjoint linear map. Prove that $V = \operatorname{ker}(T)\oplus\operatorname{im}(T)$. All I can think of is ...
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0answers
34 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = ...
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0answers
15 views

Solution verification dimension of a union of two sets

Let $V$ be a vector space. Let $W_1$ and $W_2$ be subspace of $V$. Then $W_1+W_2 = \{a+b~|~a\in W_1 \text{ and } b\in W_2\}$. (i)Prove that $W_1+W_2$ is a subspace of V. To show its a ...
1
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2answers
69 views

Find vectors when added up equal (1, 1, 1)

Question: Let $V$ be the 2-dim subspace of $\mathbb R^3$ spanned by $(1, 2, -3)$ and $(-2, 0, 1)$. Write the vector $u = (1,1,1)$ in the form $u = v + w$, where $v$ is in $V$ and $w$ is in $V^\perp$, ...
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0answers
11 views

About polynomials. If there is not two plynomials with the same grad in S, then S is linearly independent.

The problem states as follows. Let S be a set of polynomials non zero over a field F. If there is not two plynomials with the same grad in S, then S is linearly independent. I tried the following. ...
2
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0answers
22 views

computing characteristic polynomial of hyperplane arrangement

The following problem comes from Richard Stanley's $\textit{Enumerative Combinatorics}$ vol. 1, 2nd ed. It is problem 114 (c) in Chapter 3. Let $\mathcal{A}$ be a hyperplane arrangement in ...
1
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3answers
28 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
1
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2answers
24 views

Eigenvalues and eigenvectors for orthogonal projection

I've been self-teaching myself linear algebra using Treil's Linear Algebra Done Wrong and I'm currently stumped on a problem and not sure how to start it. Here is the problem: If someone could give ...
0
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0answers
9 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
0
votes
3answers
37 views

Square Root Confusion

well we know that $$\sqrt{x^2} = \pm x$$ Then if $$x^2=y^2$$ then $$\pm x= \pm y$$ Does this mean $x = y$ or $-x = -y$ or $x = -y$ or $-x = y$ or all is true? Which is true among these?
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0answers
14 views

Linear Programming problem

Find values of the variables x1, x2,, and x3 which satisfy x1 + 2x2 + x3 ≤ 16 4x1 + x2 + 3x3 ≤ 30 x1 + 4x2 + 5x3 ≤ 40 so that the minimum value of x1, x2, and x3 is as large as possible. Write this ...
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0answers
24 views

If zero is an eigenvalue of the matrix A, What does it say about A? [on hold]

An eigenvector cannot be zero, but an eigenvalue can. Suppose that zero is an eigenvalue of A. What does it say about A? Hint: One of the most important properties of a matrix is whether or not it is ...
0
votes
2answers
21 views

subspace and subspace perp dimensions equal to V

I'm trying to prove the following statement: if E is a subspace of V, then dim E + dim $E^{\perp}$ = dim V. I know this is true because when these two subspaces are added, they are equal to V, but I'm ...
0
votes
1answer
27 views

Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
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1answer
19 views

Diagonal of multidimensional DFT

If $X$ is a $n\times n$ square matrix and $F$ its Discrete Fourier Transform, is there a way to compute the diagonal $(F_{1,1},\ldots,F_{n,n})$ without explicitly computing the full DFT? How about ...
3
votes
0answers
44 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
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1answer
31 views

finding column vectors - linear transformations

$L:\mathbb{R}^3\rightarrow \mathbb{R}^2$ with bases $\mathcal{S}=\left\{\left(-1,1,0\right),\left(0,1,1\right),\left(1,0,0\right)\right\} \: \text{for} \:\mathbb{R}^3 \:\text{and} \\ ...
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0answers
22 views

linear transformation of orthogonal vector space on subspace

Let $V$ be a finite dimensional inner product space over $F$. If $W$ is a subspace of $V$, prove that the orthogonal projection of $V$ on $W$ is an idempotent linear transformation of $V$ into $W$. I ...
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1answer
23 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
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votes
1answer
20 views

Domain of compostions of linear mappings [on hold]

Let $T$ be a linear transformation from $\Bbb R^3$ into $\Bbb R^2$ and $S$ be a linear transformation from $\Bbb R^2$ into $\Bbb R^3$. Is the mapping $ST$ a linear transformation from $\Bbb R^3$ into ...
3
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0answers
25 views

Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
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votes
1answer
49 views

Dont ask - What is the relation between $f$ and $f(x)$? [on hold]

$f = a.g + b.h$, space $V$ $f(x) = a.g(x) + b.h(x)$ $f$, $g$ and $h$ are scalar valued functions. $x$ is a vector in $\Bbb R^1$ and $f$ is a vector in $V$. So $[f(x)]$ is a vector. is $[f(x)] (f) ...
0
votes
2answers
50 views

How do you solve this circular system of equations in $\mathbb{Z}_2$?

I'm trying to solve a system of equations in $\mathbb{Z}_2$ that look like this: \begin{align} x_1 + x_2 = p_1 \\ x_2 + x_3 = p_2 \\ x_3 + x_4 = p_3 \\ ... \\ x_n + x_1 = p_n \\ \end{align} I know ...
1
vote
2answers
19 views

How to find matrix of orthogonal projection from gram-schmidt orthogonalization

I'm having a little difficulty understanding Gram-Schmidt orthogonalization. I have a problem to apply Gram-Schmidt orthogonalization to the system of vectors $(1,1,1)^T, (1,2,1)^T$ then write the ...
1
vote
1answer
13 views

Showing something involving integrals is an inner product

I have this problem: Let $C([0,1])$ be the real vector space of continuous functions on the interval [0,1]. Show that $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
1
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1answer
21 views

What the limit of a matrix over time shows about the future

$x_k$ is the fraction of people who prefer cake to pie at year $k$. The remaining fraction $y_k=1-x_k$ prefer pie. At year $k+1$, $\frac{1}{5}$ of those who prefer cake change their mind. Also at year ...