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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
9 views

What's the difference between the trajectory, the phase portrait and vector field of a matrix?

Take the matrix $$\begin{pmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{3}{4} & \frac{1}{4} \end{pmatrix}$$ as an example. What's the difference between its trajectory(discrete), phase portrait ...
0
votes
2answers
10 views

matrix trace bilinear form

I'm wondering about this problem on bilinear forms : We have $\phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$ $$(A,B) \rightarrow trace(AB)$$ I've proved $\phi$ is a bilinear form ...
1
vote
0answers
8 views

SU(n) generators

What is the generalization of the Pauli matrices and Dirac matrices in higher dimensions? I am actually looking for $\sqrt{\mathbb{I}}$ but I can't use the principal root which is just $\mathbb{I}$. ...
0
votes
1answer
13 views

Find orthonormal basis of quadratic form

Q: Let $$A = \begin{pmatrix} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{pmatrix}$$ Find the quadratic form of $q: \mathbb{R}^3 \to \mathbb{R}^3$ represented by A. and find ...
0
votes
0answers
10 views

Getting The Inverse Of A Positive Definite Matrix By Mutiplying It On A Diagonal One

Is the following true ? Theorem (proved in the textbook) If $A$ is a positive definite matrix and $B$ is symmetric and of the same order then $\exists P, \Lambda: A=PP^T \space B = P\Lambda ...
3
votes
1answer
30 views

If $V_1$ and $V_2$ are vector spaces over $\mathbb{Q}$ and f is a map $f(x+y)=f(x)+f(y)$ for all $x, y$ in $V_1$. Is $f$ linear transformation?

If $V_1$ and $V_2$ are vector spaces over the field of rational numbers and $f$ is a map from $V_1$ to $V_2$ such that $f(x+y)=f(x)+f(y)$ for all $x, y$ in $V_1$ show that $f$ is a linear ...
0
votes
0answers
9 views

Characters of Representations, Composition Series and Tensor Products

Let $(\pi, V)$ be a finite-dimensional representation of $G$. Prove the following: Suppose that $(\pi, V)$ has as a composition series $\{0\} \subset V_{1} \subset \dots \subset V_{r}=V$ with the ...
0
votes
2answers
25 views

Meaning of the phrase “span of vectors contains”

What does it mean if someone says the "span of vectors" $\{(x,y,z),(a,b,c)\}$ contains the vector $(d,e,f)$? I am making up numbers here. Let me know if I need to insert actual numbers here to ...
3
votes
2answers
45 views

Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$

I need some help figuring out how to work through this problem. Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$. Sorry, forgot to ...
4
votes
4answers
123 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = ...
2
votes
2answers
20 views

Minimize Energy Function

Let $A\in\mathbb{R}^{n\times n}$ be symmetric positive definite matrix and $\mathbf{b}\in\mathbb{R}^n$. How to prove that $A\mathbf{u}=\mathbf{b}$ if and only if $\mathbf{u}$ minimizes the so-called ...
1
vote
1answer
21 views

Existence of a block upper triangular form matrix representation for a linear operator

Let $T:V \rightarrow V$ be a linear operator on a finite dimensional vector space over $F$. Let $W \subset V$ be a subspace which is $T$-invariant. Show that there exists an ordered basis ...
0
votes
1answer
16 views

Is this the (a?) correct definition for $X$ having full row rank?

Let $X$ denote a $T\times K$ matrix. I have seen the definition for full column rank as "There is no vector $c \not = 0$ with $X\cdot c = 0$. Would a definition for full row rank then be "There does ...
0
votes
1answer
46 views

Solve this system of equations without calculator

$$2a +4b +3c +5d +6e=37$$ $$4a +8b +7c +5d +2e=74$$ $$-2a -4b +3c +4d -5e=20$$ $$a +2b +2c -d +2e=26$$ $$5a -10b +4c +6d +4e=24$$ find $a,b,c,d,e$ I tried solving the system of equations above but ...
0
votes
1answer
20 views

Is $X'X$ positive definite a necessary condition for $X'X$ to have full rank?

Let $X$ be a $T \times K$ matrix. $X'X$ positive definite means that for all $c \not = 0$ $c'X'X c >0$, so then $(Xc)' Xc > 0$ which implies $(X\cdot c)\cdot (X\cdot c)$ (I'm not sure what ...
-4
votes
1answer
32 views

If all eigenvalues of A are zero then A must be similar to zero matrix. [on hold]

True or false. If true prove it else give an example.
0
votes
0answers
17 views

If $T$ is an orthogonally diagonalizable linear operator in an inner product space, show that $T^*$ is also orthogonally diagonalizable.

Suppose $V$ is an inner product space and $T$ is a linear operator that is orthogonally diagonalizable. Show that $T^*$ is also orthogonally diagonalizable. Here, $T^*$ denotes the adjoint ...
0
votes
1answer
18 views

Find upper triangular matrix C such that Cx=y

In the image above, how does one know that $c=e$ and $c$ is not equal to $f$? and $e$ is not equal to $f$? How does one know that $b=d$?
0
votes
0answers
13 views

Spectral norm of lower triangular perturbation

Suppose $A\in R^{n×n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. \begin{equation} A=I+L \end{equation} All diagonal entries of $L$ are equal to $0$, so that, ...
0
votes
0answers
15 views

Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...
0
votes
1answer
29 views

If $u$ is perpendicular to $v$ and $w$, then $u$ is perpendicular to $v + 2 w$?

True or false (give a reason if true or a counterexample if false): (a) If $u$ is perpendicular (in three dimensions) to $v$ and $w$, those vectors $v$ and $w$ are parallel. (b) If $u$ is ...
0
votes
0answers
23 views

$M_n$ is the subspace of all square matrices with trace $0$, what is the dimension of $M_n$?

There is an older post with many explanations of a more specific and less general case of a $4$ by $4$ Find the dimension of the space of $4\times 4$ real matrices with zero trace I didn't quite ...
0
votes
2answers
34 views

Linear transformation from a vector space to a field

Can anyone help me with the following question: Let $V$ be a vector space over field $F$, possibly not finite dimensional. Let $T \colon V \to F$ be a linear map. Prove that there is no subspace ...
0
votes
1answer
25 views

Showing that the following vectors are linearly independent in a subspace which they do not span.

I am trying to better understand vector spaces and dimensions. I could prove (i) via induction and the definition of linear independence? However how can I approach the questions (ii),(iii) which ...
2
votes
2answers
33 views

Row Switching Matrix

I know that there exists an elementary matrix that switches the rows in another matrix when they are multiplied, but how do you prove that this elementary matrix actually does this job? I am ...
0
votes
1answer
25 views

what are the smallest and largest values of $||v - w||$? What are the smallest and largest values of $v \cdot w$?

If $||v|| = 5$ and $||w|| = 3$, what are the smallest and largest values of $||v - w||$? What are the smallest and largest values of $v \cdot w$? How can I solve these two problems? For $||v - w||$ ...
2
votes
3answers
22 views

Proof: dimension of the vector space of solutions to the system Bx=0

I've run into a matrix dimension proof I'm having some trouble with: Let $A,B$ be $n\times n$ matrices, and let $P(A),P(B),P(AB)$ be the vector spaces of solutions to the systems ...
2
votes
1answer
20 views

Jordan canonical form in Lang's Algebra

In Lang's algebra on pp.559, he writes of the nilpotent part of a matrix $M$: "We observe also that the only case when the matrix $N$ is $0$ is when all the roots of the minimal polynomial have ...
-1
votes
1answer
27 views

Show that Pn is an (n+1)-dimensional subspace [on hold]

Show that $P_n = \{$Polynomials with real coefficients of degree $≤ n\}$ is an $(n+1)$-dimensional subspace of the infinite-dimensional vector space of all real polynomials.
3
votes
2answers
27 views

Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
0
votes
1answer
20 views

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$.

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$. In my mind, a good way to go about this proof is proving that ...
0
votes
0answers
12 views

Hyperplane of an mn-dimensional space [on hold]

Can someone explain to me why the hyperplane of a $mn$-dimensional space would have dimension $(m-1)n$?
1
vote
0answers
35 views

Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
0
votes
2answers
22 views

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$. I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking: ...
1
vote
1answer
27 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
1
vote
1answer
24 views

Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
1
vote
2answers
20 views

Finding a linear transformation with a given null space

The problem statement is, Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$. ...
0
votes
1answer
22 views

Relating regression to projection?

I recently learned that one can think of regression as a projection of a vector in a high dimension space onto the other vector. I tried implementing this and got it to work: ...
2
votes
3answers
61 views

Why do I have to show this subspace is an invariant subspace?

Consider a vector space $V \cong \mathbb{R}^n$ with an operator $I \in O(n)$ satisfying the property $I^2 = -Id_{V}$. See Linear Complex Structure for context. I want to show that $V$ has real ...
0
votes
2answers
29 views

Show that an inverse of a bijective linear map is a linear map.

So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well? enter image description here
0
votes
1answer
18 views

Dimension of the subspace of a vector space spanned by the following vectors.

I know that in order to find a subsequence that is a basis of a subspace is to check whether the given vectors are linearly independent and whether they span the subspace. However how can I find the ...
0
votes
1answer
17 views

A question concerning isometries and determinants

Let $e_1 = (1, 0)$ and $e_2 = (0, 1)$. Assume that $f : \Bbb R^2 → \Bbb R^2$ is an isometry of the plane fixing $(0, 0)$. Let $f(e_1) = (a, b)$ and $f(e_2) = (c, d)$, and let $A = \begin{vmatrix} ...
0
votes
1answer
17 views

What's the period of this matrix?

Consider the matrix $$ A = \begin{pmatrix} 0.1 & 0.3 & 0.4 & 0.2 \\ 0.2 & 0.4 & 0.0 & 0.4 \\ 0.0 & 0.3 & 0.5 & 0.2 \\ 0.5 & 0.3 & 0.2 & 0.0 ...
0
votes
2answers
21 views

Finding the basis of a subspace

I understand that the basis of a subspace defined by this equation requires you to find a combination of $x_1,x_2,x_3$ that satisfy this equation [so $(-1,0,2)$ for example]. But how do you know how ...
1
vote
2answers
24 views

Proving that if $A$ is diagonalisable then $\chi_A(A) = 0$

This could be a very simple question to answer, but I'm unsure how to prove this. If you have a diagonalisable matrix $A$, prove that $\chi_A(A)$ is the zero matrix. (where $\chi_A(x)$ is the ...
0
votes
3answers
20 views

Question about how the determinant of a square matrix can help determine whether a set of vectors is a basis.

I have a linear algebra midterm tomorrow. While it's highly unlikely a question of this type shows up, I really wanted to understand this because I am curious since I've spent so long without coming ...
1
vote
1answer
24 views

Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...
0
votes
2answers
23 views

$a+b$ for $ax+3y=5$ and $2x+by=3$

If $ax+3y=5$ and $2x+by=3$ represent the same straight line, then what does a+b equal? I've tried this, $ax+3y=5$ and $2x+by=3$ Multiply to equal 15 so they equal each other ...
1
vote
1answer
21 views

$C_o= \{(x_n):x_n \in R, x_n \rightarrow 0 \}$ and $ M=\{(x_n):x_n \in C_0,~~ x_1+ x_2+…+x_{10}=0\}$ then dimension of $(\frac{C_0}{M})$ is equal to

Let $C_o= \{(x_n):x_n \in R, x_n \rightarrow 0 \}$ and $ M=\{(x_n):x_n \in C_0,~~ x_1+ x_2+...+x_{10}=0\}$ then dimension of $(\frac{C_0}{M})$ is equal to
1
vote
1answer
11 views

Conditioning in regards to matrix vector product

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...