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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1answer
24 views

Why is the Span of a subset of a linear space defined in such at way?

If I have a subset $M$ of a linear space $E$, we define the linear span of the subset, $M$, as: $$\operatorname{span} M=\bigcap_\alpha \{E_\alpha : E_\alpha \hookrightarrow E\text{ and } M \subseteq ...
2
votes
2answers
48 views

If $A$ is a $4 \times 4$ matrix with $rank(A) = 1$, then either $A$ is diagonalizable or $A^2 = 0$, but not both

If $A$ is a $4 \times 4$ matrix with rank$(A) = 1$, then either $A$ is diagonalizable (over $C$) or $A^2 = 0$, but not both (Note that $A$ has complex entries) So far, the only thing I've tried ...
2
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0answers
34 views

Let the plane V be defined by $ax + by + cz + d = 0$; with $a, b, c, d \in \mathbb{R}$ and the vector $(a; b; c)$ a unit vector.

I am battling to get my mind around some of the concepts involving vectors in $3$-space. This question asks me whether the following statements are True or False: (A) The line $(a; b; c)$ is parallel ...
4
votes
1answer
40 views

$O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$

During a lecture of a Lie Algebras yesterday, the professor of the class stated the following fact without proof $O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$ Note that we are viewing ...
0
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1answer
20 views

parameterizing two points along a circular path

Question: Parameterize points (3, 4) to (-4, 3) along a circular path I know that if I find the vector equation for this it would be r = <3-7t, 4-t>. But I'm not sure what the question is asking ...
0
votes
2answers
54 views

Let $A$ be a $5\times 5$ matrix all of whose eigenvalues are zero. Is $A$ symmetric, anti-symmetric, or $A=-A$?

Let $A$ be a $5\times 5$ matrix all of whose eigenvalues are zero. Which of the following are always true: a. $A=-A$ b. $A^t=-A$ (anti-symmetric) c. $A^t=A$ (symmetric) d. $A^5=0$ For b: ...
1
vote
3answers
32 views

Let $A\in \mathbb C$ be a $2 \times 2$ matrix, let $f(x)=a_0+a_1x+\cdots a_nx^n$ be any polynomial over $\mathbb C$. Comment on $f(A)$

Let $A\in \mathbb C$ be a $2 \times 2$ matrix, let $f(x)=a_0+a_1x+a_2x^2+\cdots a_nx^n$ be any polynomial over $ \mathbb C$. Then which of the following is true? a) $f(A)$ can be written as ...
0
votes
1answer
16 views

Generator operator of $v$?

Let $\text{U_+}$ be an associative $\mathbb{C}$-algebra with two generators $E$, $H$, and one defining relation $HE - EH = 2E$. Let $M$ be an $\text{U}_+$-module. If $v \in M$ is a nonzero eigenvector ...
0
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0answers
7 views

Why quadratic forms with the same isotropic cone are proportional?

Let $V$ be a complex space with a quadratic forms $P$, $Q$. Assume that the isotropic cones for $P$ and $Q$ are the same: $$ P^{-1}(0)=Q^{-1}(0). $$ How to check that there is a constant $c\neq 0$ ...
1
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0answers
27 views

Diagonalization: Differential Equations

The booking being used for this course is Differential Equations and Dynamical Systems by Lawrence Perko. The problem is as follows: Let the $n\times n$ matrix $A$ have real, distinct ...
0
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1answer
28 views

using Cayley Hamilton to find a power of linear transformation

I have reached the the following formula using Cayley Hamilton: $$T^3-2T+2I=0$$ Now I need to find $T^4$ so what I did is $T(T^3-2T+2I=0)\iff T^4-2T^2+2IT=0\iff T^4=2T^2-2IT$ But the answer is ...
2
votes
2answers
37 views

Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Comment on S.

Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Then S is: one-one but not onto onto but not one one invertible non-invertible ...
1
vote
2answers
57 views

calculating the characteristic polynomial

I have the following matrix: $$A=\begin{pmatrix} -9 & 7 & 4 \\ -9 & 7 & 5\\ -8 & 6 & 2 \end{pmatrix}$$ And I need to find the characteristic polynomial so I use ...
1
vote
1answer
23 views

Uncoupled Linear System: Differential Equations

I'm trying to make sense of a problem I was given in class and I want to know if I am on the right track. The question is as follows: If $\vec{u}(t)$ and $\vec{v}(t)$ are solutions of the linear ...
0
votes
0answers
13 views

Probability that a Polynomial Has Specific Root When $y_i$'s are Not Random.

Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-1$, but degree of $P_2$ can be at most $n-1$. $P_1$ has root $\beta$, where $\beta \leftarrow ...
6
votes
3answers
117 views

Prove that $\det(I-CD)=\det(I-DC) $

Let $C$ and $D$ be matrices such that $DC$ and $CD$ are square matrices of the same dimension. How can one prove that $\det(I-CD)=\det(I-DC)$? This is my approach to the question. I am not sure ...
1
vote
0answers
16 views

Measure change/similarity between two affine transformations

I have two affine transformations $A_1$ and $A_2$ consisting only of a rotation matrix $R_i$ and a translation vector $\overrightarrow{t_i}$ (all in 3D space): $$A_i = \left[ \begin{array}{ccc|c} \, ...
3
votes
2answers
30 views

Sum of three cross products is zero.

Let $u,v,w\in \mathbb R^3$. Prove $u \times( v \times w)+v \times( w \times u)+w \times( u \times v) =0$ I guess things would work out if I just expanded as a ton of products. Is there a better way?
1
vote
1answer
28 views

Characteristic polynomial above the complex numbers

I need to find the Characteristic polynomial of $\begin{pmatrix} i+1 & 0 & 0 \\ 0 & 3i-1 & 2-2i\\ 0 & 2-2i & 3i-1 \end{pmatrix}$ I know that there is not ...
0
votes
1answer
4 views

Solving Augmented Matrix Breaking Strict Triangle Form

I'm trying to solve the following system of equations: $ 3x_1 + 2x_2 + x_3 = 0\\ -2x_1 + x_2 -x_3 = 2\\ 2x_1 - x_2 + 2x_3 = -1 $ From which I'm using the augmented matrix: $$ \left[ ...
1
vote
0answers
42 views

Valid Vector Space Proof (given v + w = 0 prove w = -v)

I'm working through Serge Lang's 'Algebra' and my answers to the proof exercise differ from his but reach the same outcome. I'm not sure if my proofs are invalid or if they are a correct alternative ...
-2
votes
1answer
35 views

how many solutions does a 3x3 linear systems have? [on hold]

I just want to know if 3x3 linear systems has no solution, one solution, or two solutions, three solutions or infinitely many.solutions
1
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2answers
31 views

proof-similar matrices have the same characteristic polynomial

I have this proof but did not understand the following step: $$ \det(xI - M^{-1} A M)= \det(M^{-1} xI M - M^{-1} A M)$$ The author said in the comments that it is due to "$xM$ commutes with the ...
1
vote
1answer
22 views

Selecting a basis such that the orientation is preserved

I need to map a polygon from a 3D plane to a 2-dimensional basis, do some processing, and project the result back to 3D. The vertices in the polygon is always ordered counterclockwise and this ...
1
vote
1answer
16 views

Determining intersection of kernel and range of a linear operator.

I was posed the following question : If T is a linear operator on vector space V and given that the kernel and range of T are disjoint, ie. they have only the zero vector in their intersection and T ...
0
votes
2answers
48 views

Proof for $V \cong V^{**}$

Theorem: Let $V$ be an vector space. Then the dual space of $V$'s dual space is canonically isomorphic to $V$. I am able to prove that $V$ is a subspace of $V^{**}$, the map ...
1
vote
1answer
21 views

How to find a $3\times 3$ matrix with a certain null space and certain column space

We were asked to Find a $3\times 3$ matrix whose null space is the $x$-axis and whose column space is the $yz$-plane. I was told that the answer is this, but I didn't understand why: ...
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votes
0answers
21 views

In vector space why we not take 1.a=a.1 [on hold]

I solve many question which satisfies this condition.I want to answer this
0
votes
1answer
25 views

Is it possible for a matrix to have nullity different from its transpose?

Say I have a $2\times 3$ matrix with $\operatorname{rank}(A)=1$ and $\operatorname{nullity}(A)=2$, i.e. only one pivot point and two free variables in the solution set of this system because we have ...
2
votes
3answers
66 views

If square matrix A satisfying $A^2-4A+4I=0$ does it follow that A is diagonizable?

I am given the following statement and asked to determine whether it is true or false: If A is a n x n matrix, and $A^2-4A+4I=0$, then A is diagonizable. Any help is appreciated, thank you.
3
votes
1answer
29 views

How many numbers between 1 and 10000, inclusive, are multiples of 12 or 20?

I calculated the multiples of 12 and multiples of 20, 833 and 500 respectively. Now I calculated the multiples of 12 * 20 = 240,and as a result have 41. The solution would be 833 + 500-41 = 1292 ...
0
votes
2answers
32 views

Show that the full null space of the matrix A and its column space in the plane 2x+2y - z = 0

Show that the full null space of the matrix A = $\begin{bmatrix} 0&1&5\\ 1&0&0 \\ 2&2&10 \end{bmatrix}$ is the line $\lambda$(0.-5,1), $\lambda \in \mathbb R^3$ and its ...
3
votes
0answers
29 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = ...
0
votes
1answer
22 views

about scaling property of proximal operator

If the proximal operator of $f(x)$ is $\text{prox}_{\lambda f}(x)$, what about $cf(x)$ and $f(cx)$, c is a scalar. For example, If $f(x) = ||x||_{1}$, $x \in \mathbb{R}^{n}$, how about the proximal ...
2
votes
0answers
28 views

Linear map is diagonalizable iff its adjoint is diagonalizable

Problem Let $V$ be a finite inner product space and let $T:V \to V$ be a linear transformation. Prove that $T$ is diagonalizable if and only if the adjoint transformation $T^{*}$ is diagonalizable. ...
2
votes
2answers
38 views

Index notation interpretation for matrices

I want to understand the how to interpret the matrices which are represented by index notation. Here is my matrix $𝜎_{𝑖𝑗}+𝜎_{π‘–π‘˜}𝑀_{π‘˜π‘—}βˆ’π‘€_{π‘–π‘˜} 𝜎_{π‘˜π‘—}$ All the matrices in the equation ...
1
vote
2answers
36 views

Explicit example of a basis of invertibles for $n\times n$ matrices

Using a topological (+linear algebra) argument, one can establish the existence of a basis spanning any square matrix using invertible matrices ( $span(GL_n (\Bbb{R}))=\mathcal{M}_n (\Bbb{R}) $). But ...
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0answers
33 views

Least squares solutions of the linear system

I'm doing problems from old exams, and my solutions don't add up with the professor's solution. The problem is as followed: Find all least squares solutions of the linear system. I checked my ...
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0answers
20 views

mathematics representation on operators [on hold]

(a) Let A be an operator on C and |1i = 2 which has the following action on the canonical basis vectors |0i = 0 : 1 A|0i = 2|0i + 3|1i A|1i = 1|0i βˆ’ 4|1i. (i) Find the matrix representation of ...
2
votes
1answer
37 views

Orthogonality v. Perpendicularity

In Intro to Linear Algebra (my class) two vectors are defined to be orthogonal if their dot product is zero. And the dot product of two $n$-vectors $\vec a\cdot\vec b=0$ means that the two vectors are ...
1
vote
1answer
49 views

Intuition: why distinct eigenvalues -> linearly independent eigenvectors?

Suppose you have an n x n matrix with n distinct (not repeated) eigenvalues. There is a theorem telling us that the eigenvectors corresponding to these eigenvalues must be linearly independent. I can ...
0
votes
0answers
13 views

Derivating the normal vector from a line equation

$$g_a: \binom{x}{y}=\binom{x_g}{y_g}+\lambda\binom{a_{g_a}}{b_{g_a}}$$ $$g_b: a_{g_b}x+b_{g_b}y=c$$ Explain why: $$\vec{n_{g_a}}=\binom{a_{g_b}}{b_{g_b}}$$ Where ...
0
votes
3answers
33 views

matrix times its transpose equals minus identity

What would be a good example for a $n\times n$ matrix such that $A^{T}A=-I$? It would be better if you can give a matrix which has a well-known name (like "rotation matrix" etc). Thanks!
0
votes
1answer
34 views

Find a $4\times 3$ matrix to make a $3 \times 4$ matrix invertible [duplicate]

The matrix $P$ is 3x4 given. Is there a 4x3 matrix $Q$ such that the product of $QP$ is invertible? If this can't happen can someone explain why?Thanks. Matrix P
1
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3answers
56 views

Can the product of an $4\times 3$ matrix and a $3\times 4$ matrix be invertible?

I want to find the inverse of the product of $2$ non-square matrices, but is this even possible?
0
votes
0answers
25 views

Find a relation between a,b and c

$ a,b,c\in \Bbb R$ $2x_1+2x_2+3x_3=a$ $3x_1-x_2+5x_3=b$ $x_1-3x_2+2x_3=c$ if a,b and c is a solution of this linear equation system find the relation between a,b and c I dont understand the ...
2
votes
2answers
41 views

Computing matrix exponential of non-diagonalizable 2x2 matrix

Compute $e^M$ where $M=\begin{bmatrix}8 & -1\\4 & 4\end{bmatrix}$ Because M is not diagonalizable i try to use Jordan decomposition so i find the Jordan matrix to be $J=\begin{bmatrix}6 & ...
1
vote
0answers
27 views

Eigenvalues of Moore–Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ...
0
votes
1answer
25 views

Union of subspace

Q. Say U and W are subspaces of a a finite dimensional vector space V (over the field of real numbers). Let S be the set-theoretical union of U and W. Which of the following statements is true: a) ...
2
votes
1answer
56 views

For Which Value The Matrix is Diagonalizable?

For which values of $a$ the matrix $\left(\begin{array}{ccc} 2 & 0 & 0 \\ 2 & 2 & a \\ 2 & 2 & 2 \end{array}\right)$ is diagonalizable: above $\mathbb{R}$ above ...