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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Linear Algebra free variables

I have started taking a basic course in linear algebra and have a doubt. I understand that in order to know if AX=B Has solutions and how many of them..first we modify the matrix to its echelon form. ...
3
votes
1answer
93 views

Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?

Let $A \subset M_7(\mathbb{Z}_2)$ be a subring such that no proper nonzero subgroup $V \subset \mathbb{Z}_2^7$ is invariant under all matrices in $A$. I suspect that $A \cong \mathbb{F}_{128}$, but ...
4
votes
5answers
235 views

Intuition for the Cauchy-Schwarz inequality

I'm not looking for a mathematical proof; I'm looking for a visual one. I'm having trouble understanding (in my mind's eye) why the dot product of two vectors V and W produces a scalar that is less ...
1
vote
1answer
16 views

Deriving simple linear regression from normal equations

The normal equations of least squares regression $$X \beta = Y$$ yields the solution $\beta = (X X^T)^{-1} X^T Y$. For simple (1-dimensional) regression, the solution to $\beta x_i + \alpha = y_i$ is ...
0
votes
2answers
72 views

Why is $\{1+2x-x^2,4-2x+x^2,-1+18x-9x^2\}$ not a basis on $\Bbb P_2(\Bbb R)$?

Why is $\{1+2x-x^2,4-2x+x^2,-1+18x-9x^2\}$ not a basis on $\Bbb P_2(\Bbb R)$? The set contains three vectors, and they do not appear to be linearly dependent of each other,
1
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1answer
34 views

Find a Basis $B$ of $R^2$ so that $B$ matrix of $T$ is diagonal

$T([1,1]^t) = [3,7]^t$ $T([1,-1]^t) = [1,1]^t$ Here's what I get: $T= \left(\begin{array}{cc}3 & 1 \\7 & 1\end{array}\right) $ The eigenvectors of $T$ is $E = \left(\begin{array}{cc} .4798 ...
1
vote
1answer
30 views

Different formulas for matrix transformations

I am a bit confused about how to get a matrix in a new basis. On the one hand, we always use the multiplication by transformation matrix when we want to receive a matrix in a new basis: $A' = CA$, ...
1
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0answers
37 views

How to solve the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ in Matlab when nullitity$(A)\neq 0$

Say, $A= \begin{pmatrix} 1 & 0 &1 \\ 0 & 1 &1 \\ 0 &0 &0 \end{pmatrix}$ and $\overrightarrow{b}= \begin{pmatrix} 8 \\ -5 \\ 0 \end{pmatrix}$ and I want to solve ...
0
votes
0answers
33 views

Commutator subgroup of general linear group [on hold]

Let $G$ and $S$ be the group of all invertible $n\times n$ matrices and invertible matrices with determinant $1$ of the same order respectively over the field of real numbers. Prove that $S$ is ...
0
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2answers
25 views

The Matrix of $T$ Relative to the Ordered Bases $\mathcal B$ and $\mathcal B'$ for $V$

Let $V$ be an $n$-dimensional vector space over a field $F$, let $\mathcal B$ and $\mathcal B'$ be different ordered bases for $V$, and let $T$ be a linear operator on $V$. Then Hoffman and Kunze in ...
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1answer
52 views

$Hom(V,W)$ remains unchanged when norms of $V$ and $W$ are replaced with equivalent norms.

I was thinking about the following question from section 3.4 of Loomis and Sternberg's Advanced Calculus The fact that $Hom(V,W)$ is unchanged when norms are replaced by equivalent norms can be ...
0
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2answers
25 views

co-ordinate geometry question 2

Find the equation of the line which cuts off an intercept 4 from x-axis and makes an angle 30 with y-axis . Find the equation of the line joining points (-1,3) and (4,-2) ?
5
votes
1answer
67 views

Prove that $A,B$ are similar

Let $A,B\in M_6(\mathbb{Q})$, such that the minimal polynomials are $$m_A(x) = m_B(x) = x^2-x-1$$ Prove that above $\mathbb{Q}$, $A$ and $B$ are similar. So $x^2-x-1$ is of course ...
2
votes
3answers
31 views

co-ordinate geometry question

Find the equation of the lines which passes through the point (3,4) and whose intercept on y-axis is twice that of x-axis ?
5
votes
5answers
553 views

Why the determinant of a matrix with the sum of each row's elements equal 0 is 0?

I'm trying to understand the proof of a problem, but I'm stuck. In my book they consider that if all lines of a matrix has sum 0 then it's determinant is also 0. I checked some random examples and ...
1
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0answers
25 views

Help me identify whether given linear equations are homogeneous

To sum up what I'm stuck with Let $R$ be a $m\times n$ row-reduced echelon matrix and $W$ be the row space of $R$. Suppose that $p_{1},...,p_{r}$ are the non-zero row vectors of $R$ and the first ...
0
votes
1answer
64 views

Vector inequation problem

$${A_i} = \left( {\begin{array}{*{20}{c}}{{A_{i1}}}\\{{A_{i2}}}\\ \vdots \\{{A_{in}}}\end{array}} \right),{B_i} = \left( {\begin{array}{*{20}{c}}{{B_{i1}}}\\{{B_{i2}}}\\ \vdots ...
2
votes
2answers
62 views

Conditions for an orthogonal matrix equation

Let $B_1$ and $B_2$ be given $n \times n$ real non-singular matrices and consider the system of equations $$\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\begin{bmatrix}B_1 ...
0
votes
0answers
24 views

Describe the Jordan Normal Form of this operator,

In a previous question on MSE, I computed a 15x15 matrix of an operator. We see that the operator is nilpotent, with spectrum = {0}. But the last part of the problem asks to describe the Jordan ...
2
votes
2answers
43 views

How to prove the existence of vectors?

I solved this problem for a few specific vectors but I don't know how to prove this in general? Grateful for any kind of help!
1
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0answers
34 views

Basis, Ordered Basis, and Linear Independence

I am given $$ \mathscr B=\left\{\begin{pmatrix}1\\1\\0\\0\end{pmatrix},\begin{pmatrix}1\\1\\1\\1\end{pmatrix},\begin{pmatrix}1\\1\\1\\1\end{pmatrix},\begin{pmatrix}1\\0\\0\\0\end{pmatrix}\right\} $$ ...
4
votes
1answer
114 views

Question concerning the group over GL$(n,\mathbb{Z})$

Is every vector $[a_1,a_2,\dots, a_n]$ with $\gcd(a_1,a_2,\dots,a_n)=1$ a column in some matrix $A\in GL(n,\mathbb{Z})$? I don't think this is a duplicate: Let me rephrase this questions using ...
4
votes
2answers
88 views

How can I tell that my matrix is nilpotent?

I just computed a 15x15 matrix by hand :( It is not upper triangular as I hoped it would be. But my computations agree with what's offered in the student solution. My question is: the solution ...
1
vote
2answers
23 views

What exactly is the reason as to why the solution space of the characteristic equation of a matrix give the eigenvalues of that matrix?

After studying some linear algebra, i've learned about how the $\lambda$'s satisfying det$(A-\lambda I)=0$ are eigenvalues of $A$. But I am failing to see the connection as to why. Why is this?
0
votes
2answers
49 views

Condition for vector to have redundant coordinates

Consider the vector $V\in\mathbb{R}^6$: $$V=\begin{bmatrix} -1&1&-1&-1 \\ -1&1&1&-1 \\ -1&-1&1&-1 \\ -1&1&1&1 \\ -1&-1&1&1 \\ ...
1
vote
1answer
24 views

Replacing pinv with inv in MATLAB

Let $\mathbf{y} = \mathbf{Ax}$ represent a system of equations where $A\in\mathbb{R}^{m\times m}, x\in\mathbb{R}^{m\times 1}$. However rank of $\mathbf{A}$ is $m-1$. I add another equation ...
0
votes
1answer
56 views

Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$ [on hold]

$Z_1,Z_2$ and $Z_3$ are affixes of points of equilateral triangle $ M_1 ,M_2$ and $M_3$. Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$.
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41 views

What is the sum of these subspaces? [on hold]

Suppose that \begin{align*} U&= \{(x,x,y,y)\in\Bbb F^4:x,y\in\Bbb F\} & W&= \{(x,x,x,y)\in\Bbb F^4:x,y\in\Bbb F\} \end{align*} In the textbook the rest of the problem states: Then U + W ...
0
votes
0answers
26 views

Legendre transformation is symmetric

Let f:$\mathbb{R}^n \to \mathbb{R}\cup \{ \infty \}$ be proper convex lower semicontinuous $\leftrightarrow$ Epi(f):={(x,t):f(x)$\leq$t} is nonempty, convex, and closed. Its Legendre transformation is ...
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0answers
23 views

What is subordinate matrix norm?

What is 'subordinate matrix norm' in this question? .
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0answers
12 views

Simplification of pseudo inverse of skinny fat matrix in least squares

In solving a least squares problem, I have a pseudo-inverse in the regular form $(A^TA)^{-1}A^T$ where $A = \left[\begin{array}{c} I \\ \sqrt{\lambda} B \end{array} \right]$. The solution claims that ...
0
votes
1answer
30 views

How to prove the following condition on elements of a positive semi definite matrix?

Given: $2$x$2$ positive semi definite matrix $T$ over complex field with real elements. That is for every vector $v \in \mathbb{C}^2$, $v^T Tv \ge 0$ and $T_{ij} \in \mathbb{R}, 1 \le i,j \le 2$ . ...
0
votes
0answers
40 views

Wondering how to rotate a normal vector in 4 dimensions?

Saw another post that suggested a answer but need help with the answer and the other post is inactive. I know how to rotate in 3-space using matrix transforms for each axis no problem. Have a very ...
0
votes
0answers
34 views

Given two points, $x_1$ and $x_2$, find $x$ such as $ (x-x_1)$ is a projection into $(x_2-x_1)$ at $(x_2-x_1)/2$

Given two n-dimensional points $x_1, x_2$ that creates a vector $v = x_2 - x_1$. I want to find a point $x$ such as $u = x-x_1$ projects onto $v$ into the point $0.5(x_2-x_1)$. In other words, find $x ...
3
votes
2answers
87 views

Prove that $V = \ker T \oplus \text{Im}T$

Let $T:V\to V$ such that $f_T = \sum_{i=0}^n c_ix^i$ and $c_1 = c_n = 1, c_0 = 0$. Prove that $V = \ker T \oplus \text{Im}T$. My thoughts so far: For some basis $B$, we have $[T]_B = A$. We know ...
0
votes
0answers
18 views

SVD of partitioned matrix where all cells except one are zero

Let $A$ be a real valued matrix of size $n \times n$. Let the SVD of $A$ be $$A= UDV^T.$$ I am interested in $$Q=VU^T.$$ Now assume we expand $A$ with zero rows and columns to get the block matrix ...
1
vote
3answers
60 views

find the Jordan form and $P$ such that $P^{-1}AP = J$.

Consider the matrix $$A = \left(\begin{array}{cccc} -11&0&-9\\32&1&24\\16&0&13 \end{array}\right)$$ I want to find the Jordan form of $A$, with $1$-s at the bottom and the ...
1
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0answers
18 views

Simultaneous triangular form for nil algebras over division ring

Let $D$ be a division ring. Let $V$ be a finite dimensional module over $D$, let $I \subseteq\operatorname{End}_D(V)$ be a $D$-submodule on both sides (I mean a subgroup closed by both on left and ...
1
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1answer
27 views

Shortcuts for computing the eigenvalues of a linear transformation

How would you calculate the eigenvalues of the following matrix? $A = \begin{pmatrix} -3 & 1 & -1 \\ -7 & 5 & -1\\ -6 & 6 & -2\end{pmatrix}$ $ $ $\ \ \ \ \ $$\chi_A(\lambda) ...
0
votes
1answer
33 views

Question regarding Cramer's rule proof

I understand that Cramer's rule can be shown from: $$Ax=b\iff x=A^{-1}b\iff x=\frac{adj(A)}{det(A)}\cdot b$$ I do understand that $adj(A)\cdot b$ is equal to calculating the determinant when ...
1
vote
2answers
57 views

Eigenvalue Problem — prove eigenvalue for $A^2 + I$

This is a proof I've been trying to figure out since the problem was presented to me. We are given that $\lambda$ is an eigenvalue for a matrix $A$ and the vector $u$ is the eigenvector corresponding ...
1
vote
1answer
55 views

Prove eigenvalue for $A^2 + I$ [duplicate]

This is a proof I've been trying to figure out since the problem was presented to me. We are given that $\lambda$ is an eigenvalue for a matrix $A$ and the vector $u$ is the eigenvector corresponding ...
3
votes
2answers
47 views

QR decomposition of smooth function

I need some help on this problem. Suppose $A(t)$ is a smooth family of $n\times n$ real matrices with $A(0)=I$ and write $A'(t)$ as derivative of $A$. (a) For each $t$, write, $A(t)=Q(t)R(t)$ where ...
0
votes
2answers
48 views

$A v = \lambda v \implies A^* v = \bar{\lambda} v$ if $A$ is normal [duplicate]

I want to show that if $A$ is normal then $$ A v = \lambda v \implies A^* v = \bar{\lambda} v $$ I can show that $A^*v$ is also an eigenvector of $A$, using the fact that $A$ and $A^*$ commute, but ...
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0answers
26 views

Cayley Hamilton Theorem using LU decompostion

I am trying to find the characterisitic equation of n*n matrix by Cayley Hamilton Theorem using LU Decompostion. Below is my algorithm to find U matrix. ...
0
votes
1answer
18 views

Cremar rule-geometric interpretation

I have been reading the information about this on Wikipedia all was clear until this sentence: "Now, this last parallelogram, by Cavalieri's principle, has the same area as the parallelogram ...
0
votes
1answer
22 views

Application of the Chinese Remainder Theorem for polynomials

Given the polynomials $g(t) = t$ and $h(t) = (t-3)^2 \in \mathbb{C}[t]$, I want to find the smallest (in terms of degree) polynomial $f(t) \in \mathbb{C}$ satisfying $f \equiv 0$ mod $g$ and $f \equiv ...
3
votes
1answer
35 views

Cramer Rule Over Finite Field

Let $A=\pmatrix{4&2\\ 0&1},\ b=\pmatrix{5\\ 3}$ and $A\pmatrix{x_1\\ x_2}=b$ over the field $\mathbb Z_7$. What is $x_1$? So we need to calculate $$x_1=\frac{\det(A_1)}{\det(A)}$$ ...
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votes
0answers
31 views

I have to show $A$ commute with a non-central matrix whose nullity is at least $\dfrac {(p+1)}{2}$. [on hold]

Let $F$ be a field and $p\geq 3$ is a prime number, and $A \in M_p(F)-FI$ is a non cyclic matrix, then using the rational form of $A$. I have to show $A$ commute with a non-central matrix whose ...
1
vote
3answers
70 views

Matrix Exponential and Logarithm

Consider the following matrix $A$: $A = \begin{bmatrix} \cos^2(1) & -\sin(2) & \sin^2(1) \\ \cos(1)\sin(1) & \cos(2) & -\cos(1)\sin(1) \\ \sin^2(1) & \sin(2) & ...