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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1answer
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Determinant of a matrix and chech whether it is non negative definite or not

Let $V = \{ f : [0,1] \to \mathbb R | f$ is a polynomial of degree less than or equal to n $\}$. Let $f_j(x) = x^j$ for $0\leq j \leq n$ and let $A$ be the $(n+1) \times (n+1)$ given by $a_{ij} = \...
5
votes
2answers
104 views

Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
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1answer
32 views

Exercise on three planes meeting in a line.

In $R^3$, Given the plane $\pi : ax + by +cz + d = 0$ and the planes $\alpha : y + z = 2, \quad \beta: x - y + z = 0$ . Do there exist values of $a,b,c,d$ s.t. the three planes meet two by two in a ...
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0answers
25 views

Prove $P(\mathbb{R})\subset C(\mathbb{R}))$ is dependent, $ C(\mathbb{R}))$ contains all continous functions

I know I can represent a polynomial as: $$P_n(\mathbb{R})=a_0 x^{n} + a_1 x^{n-1} + \dots + a_n$$ I want to find if this has only one solution or not: $$C_1P_1(x)+C_2P_2(x)+\dots+C_nP_n(x)=0$$ From ...
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0answers
20 views

Matrices in $F_n^m$ proof

Let F be a field and X, Y two F-linear subspaces with $dim_FX=n\in \mathbb{N}$ and $dim_FY=m\in \mathbb{N}$. To show: a)There exists a matrix $A\in F_n^m$ with $def(A)=0 \iff n\leq m$ b)There ...
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0answers
36 views

If $T\colon \mathbb{R}^n \to \mathbb{R}^n$ is linear and $E$ is measurable then $T(E)=\lvert \det{T}\rvert\cdot \lvert E\rvert$

Let $T\colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear mapping and let $E$ be a measurable set in $\mathbb{R}^n$. Show that $$\lvert T(E)\rvert=\lvert \det{T}\rvert\cdot \lvert E\rvert.$$ I have ...
2
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3answers
46 views

Show that if A is self-adjoint and $A^{2}x=0$, show that $Ax=0$.

I feel like i'm overcomplicating this a bit. Let $X$ be a finite-dimensional inner product space and $A$ be a linear transformation from $X$ to $X$. If A is self-adjoint and if $A^{2}x=0$, show that $...
0
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1answer
27 views

Building a surjective function

Let F be a field and X, Y two F-linear subspaces with $dim_FX=n\in \mathbb{N}$ and $dim_FY=m\in \mathbb{N}$. Show that: There is a surjective F-linear mapping f:X$\rightarrow Y \iff n\geq m$ I have ...
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2answers
67 views

Points on which function is invertible?

$f: \mathbb R ^{2}\mapsto \mathbb R ^{2}$ $f(x,y)\mapsto((x-y)^{2}+1, x-y^{3}-2)$ For which points is this function invertible? I calculated the Jacobian matrix, but what should I do next to get ...
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2answers
51 views

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$ Basically if I have found the orthonormal basis for the span of S can I use that to find the dimension ...
0
votes
1answer
22 views

Property of an invertible matrix that row reduced form is identity matrix

If a the row reduced form of a $n \times n$ matrix is the equivalent $n \times n$ identity matrix. Is the $n \times n$ matrix always invertible? Furthermore if the row reduced form is not the ...
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0answers
45 views

I am confused about this notation for matrix representation.

I am confused about this notation from the image below. How can you represent something like (1,2),1 as a 2D matrix? For S1 = (0,1,2) and S2 = (0,1) I would expect two matrices like: [(1,0),(1,0),...
3
votes
1answer
41 views

$(AB=BA\wedge A^*Bx=0)\implies BA^*x=0$?

Let $X^*$ mean the conjugate transpose of matrix $X.$ I am given two matrices $A,B$ and a vector $x$ such that $AB=BA$ and $A^*Bx=0.$ Does $BA^*x=0$ then? It may look out of context, but such ...
0
votes
2answers
37 views

Finding the action of T on a general polynomial given a basis

I am given a question as follows: Suppose $T: P_{2} \rightarrow M_{2,2}$ is a linear transformation whose action on a basis for $P_{2}$ is $$T(2x^2+2x+2)=\begin{bmatrix} 2&4 \\ 2&8 \end{...
1
vote
1answer
43 views

If $A$ and $B$ have the same degree of nilpotence, do they have the same rank?

Let $A, B$ be nilpotent $n\times n$ matrices over the field $K$. Is the following correct? If $A$ and $B$ has the same degree of nilpotency, then $\operatorname{rank} A = \operatorname{rank} B $
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6answers
84 views

How to determine which of the following transformations are linear transformations?

Determine which of the following transformations are linear transformations A. The transformation $T_1$ defined by $T_1(x_1,x_2,x_3)=(x_1,0,x_3)$ B. The transformation $T_2$ defined by $T_2(...
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votes
3answers
49 views

How to find the images of u and v under T?

Let $A = \begin{bmatrix} -1 & -4 & -8 \\ 8 & -7 & 4 \\ \end{bmatrix}_.$ Define the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^2$ by $T(x) = Ax$. Let $u = \begin{bmatrix} -...
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0answers
15 views

Find an example of a lattice such that LLL algorithm can't find the shortest vector of the lattice, satisfying…

I want to find an example of a basis of a lattice of dimension $n$ such that LLL algorithm can't find the shortest vector of the lattice, and such that the shortest vector of this lattice, say $b=...
1
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3answers
29 views

Evaluate integral over path using parametrisation

Evaluate the integral of $(F.dr)$ over the path (0,0,0) -> (1,1,1) where $F=e^{-x}i +e^{-y}j + e^{-z}k$ using parametrisation [x=t, y=t, z=t] I know from simpler questions in class that you must find ...
0
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2answers
28 views

Let u,v,w be three linearly independent vectors in a 7x1 matrix, determine a value of k

Let $u,v,w$ be three linearly independent vectors in $\mathbb{R}^7$ , determine a value of $k$ $k =$ ________ , so that the set $S = {u -5v, v-3w, w-ku} $ is linearly dependent I'm very ...
2
votes
3answers
30 views

How to find specific variables that cause vectors to be linearly independent / dependent

The vectors $v= \begin{bmatrix} 5\\ 2\\ 7\\ \end{bmatrix}, u = \begin{bmatrix} 4\\ 4\\ 13+k\\ \end{bmatrix}, \text{and } w = \begin{bmatrix} -4\\ -2\\ -6\\ \end{bmatrix} $ are linearly independent ...
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3answers
34 views

Which of the following sets of vectors are linearly independent?

Which of the following set of vectors are linearly independent? I'm a bit confused. I think the answer(s) would be A, C, and D. I'm unsure of how to actually figure out if each one is or isn'...
1
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1answer
65 views

Show that $\langle x,y\rangle_A = \langle Ax,Ay\rangle$ is an inner product on $\mathbb R^n$

Let $A$ be an $n \times n$ matrix with real enteries. Define $\langle x,y\rangle_A = \langle Ax,Ay\rangle, \quad x,y \in \mathbb R^n$ , where $\langle,\rangle$ is a standard inner product on $\mathbb ...
0
votes
1answer
22 views

Trying to derive an equation to express number of carps in play

I play magic the gathering. This is the situation, without needing to know the rules: You gain one total available "mana" per turn. So available mana = number of turns passed. Your only creature, of ...
4
votes
5answers
385 views

Determine matrix of linear map

Linear map is given through: $\phi\begin{pmatrix} 3 \\ -2 \end{pmatrix} =\begin{pmatrix} -3 \\ -14 \end{pmatrix} $ $\phi\begin{pmatrix} 3 \\ 0 \end{pmatrix} =\begin{pmatrix} -9 \\ -6 \end{pmatrix}$ ...
2
votes
3answers
68 views

Find equation for line segment

I already know that the standard equation for a line is $y=mx+b$, but what if i want the line to have specific endpoints and not go on forever. For example the equation for a line beginning at $(3,1)$ ...
0
votes
1answer
50 views

Prove the orthogonal matrix with determinant 1 is a rotation

Let's define "preserve orientation" in the following way (I am not sure it is right, pls point out if there is something wrong): For a linear transformation, we only need to check non-parallel ...
2
votes
0answers
32 views

What is the name of this problem? linear Matrix equation optimization?!

I have almost no knowledge in linear algebra but I need to understand the process of solving a problem. In fact I'm looking for some keywords or hints to know what exactly should I be Googling! So any ...
2
votes
3answers
85 views

Why can't a set of four vectors in $\mathbb{R}^3$ be linearly independent?

Why can't a set of four vectors in $\mathbb{R}^3$ be linearly independent? I know that if the determinant of the vectors together is not $0$ then the vectors are linearly independent. But this is ...
1
vote
3answers
56 views

Matrix with orthonormal base [on hold]

I have the two following given vectors: $\vec{v_{1} }=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ $\vec{v_{2} }=\begin{pmatrix} 3 \\ 0 \\ -3 \end{pmatrix} $ I have to calculate matrix $B$ so that ...
0
votes
2answers
76 views

If $X$ is zero matrix,what is $e^X$?

Let $X$ be an n×n real or complex matrix. The exponential of $X$, denoted by $e^X$, is the n×n matrix given by the power series $e^X =\sum_{k=0}^{\infty} X^k/k!$ where $X^{0}$ is defined to be the ...
0
votes
1answer
22 views

Associative law of scalar multiplication of a vector

If a,b are two real numbers and v is a vector, prove that: a*(bv) = (ab)*v. I am trying to prove this equality by not using the fact that a vector can be represented by coordinates, but just by ...
7
votes
1answer
80 views

What does the word “norm” stands for in linear algebra?

I know that "norm" is the formal name for length, but where did this name came from? or from what language is came from? Thank you in advance.
2
votes
2answers
52 views

How to find the restrictions of side length on an obtuse triangle

Question: In Triangle ABC, the angle ∠ABC is an obtuse angle. The Side AB is 1cm, and the side BC is 3cm. Side AC is (3x+10)/(x+3) cm Find the restriction(s) on x. I have tried a few different ...
0
votes
1answer
38 views

Solve $a-1.73d=0, b-1.73d=0, c-1.73d=0, a+b+c -1.73d=0$ [on hold]

How can we find nontrivial solutions of the homogeneous equation $$a-1.73d=0, b-1.73d=0, c-1.73d=0, a+b+c -1.73d=0$$ I need to find the values of $a,b,c$ and $d$. When I tried with Gauss ...
2
votes
1answer
25 views

Two real symmetric matrices are congruent if and only if they have the same rank and signature.

So I saw this statement in an exercise : Two real $n \times n$ matrices are congruent if and only if they have the same rank and the same signature. But I was wondering why do we need to state ...
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3answers
32 views

Properties of RREF 3x3 matrix is the identity

The row reduced echelon form of a 3 × 3 matrix A is the identity. State whether each of the following is true or false. You do not need to explain your answers. (a) A has an inverse. (b) The columns ...
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0answers
37 views

What is the subspace of the particular solution to $Ax = b$?

If I solve the equation $$ \begin{bmatrix}1 & 2 & 1 \\ 3 & 2 & -1 \\ -1 & 2 & 3\end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} = \begin{bmatrix}0 \\ 4 \\ -4\end{...
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2answers
38 views

Computing the span

Compute the span of $\begin{bmatrix} 4\\2\\10 \end{bmatrix}$ and $\begin{bmatrix} 6\\3\\15 \end{bmatrix}$ I just don't even understand what "Compute the span" is even asking me. Can anyone give me a ...
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2answers
41 views

show that Row(A) is perpendicular to Null(A)

Hi can you please help me check my work Question: Prove that if A is a m x n matrix, vector x is an element of Row(A) and vector y is an element of Null(A), then vector x is perpendicular to y. ...
0
votes
3answers
53 views

How do you compute the span of a 3x1 matrix?

How do you compute the span of a 3x1 matrix? for Example: Compute the span of $\begin{bmatrix} 4\\0\\1 \end{bmatrix}$ and $\begin{bmatrix} 1\\0\\4 \end{bmatrix}$
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2answers
31 views

Write the vectors u, v, w, z in terms of a and b.

Write the vectors $u, v, w, z$ in terms of $a$ and $b$. I'm unsure of how to do this.. If someone could give me an example of one being done I'm almost positive I could mimic it and figure out the ...
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0answers
41 views

Incorrect answer - Simultaneous Differential Equations

The questions states solve for y such that $$y' = \begin{bmatrix} -4 & 2 & 1 \\ 1 & -3 & 1 \\ 3 & -3 & -2 \\ \end{bmatrix}y , y(0)= c = \begin{bmatrix} 1\\5\\3 \end{...
0
votes
1answer
45 views

Prove set of 5 elements of $M_{2\times 2}$ is linear independent

I want to prove if a set of $5$ elements of $M_{2\times 2}$ is linearly independent. Since I have $5$ elements I think it's impossible for it to be independent since when I sum up these matrices I'll ...
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votes
1answer
59 views

what are the possible values of dim(Null(L))?

can you please explain this question to me? Thanks Question : Suppose that L: R^4 ----> R^2 is a linear transformation. a) what are the possible values of dim(Null(L))? b) For each possible value ...
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1answer
32 views

Help with proving a 2 by 2 determinant is the area of parallelogram

I have proved a large part of this by the following but get stuck at the last step. To say $A=ad-bc$, we still need $ad>bc$. I have puzzling over this for hours. Thank you!
0
votes
2answers
28 views

Proving $(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$ , $P$ idempotent matrix.

Given that a matrix $P$ is idempotent how to prove the following relation: $$(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$$ $c$ is any real constant.
1
vote
1answer
24 views

Collinear Points in 3-Dimensions

The points A(3, -1, z), B(1, 2, 6), and C(x, 8, 14) are collinear. Find the values of x and z. I have tried finding common ratios between the points, but no common ratio is possible, I have a feeling ...
1
vote
3answers
52 views

Method of Proof in Showing Something is Smallest (Subspace)

I am reading a proof that shows the sum of subspaces is the smallest subpsace containing all the summands (It is a vector space over $\mathbb{R^n}$). The author of the book goes to show first it is a ...
0
votes
1answer
27 views

Power method and convergence

I am working on some practice problems for the convergence of power method for some given recursion relationship and I am trying to generalize/reflect on the question after having been stuck on the ...