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

0
votes
1answer
141 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
2
votes
2answers
160 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
1
vote
1answer
25 views

Can we list down all order 4 integer valued 3 x 3 matrices

Can we list down all integer 3 x 3 matrices($A$) whose are order 4 i.e $A^4= I$? or atleast get some examples? What should be the method for such thing?
1
vote
3answers
213 views

Creating and solving large systems of equations

I am trying to follow a solution in a book so that I can build my own model. They produce the set of equations below. The book claims it to be a system of equations with 10 unknowns; however from my ...
1
vote
1answer
57 views

null space of an n-by-m matrix

I have an $n$-by-$m$ ($n>m$) matrix named $J$. I wanted to find its null space so as I used matrix $M$ defined bellow: $$JM=0\text{, when } M=I-J^\dagger J$$ $J^\dagger$ is the pseudo inverse of ...
0
votes
0answers
45 views

Calculus and Matrices

Suppose I have a linear operator $T: \mathbb{R} \rightarrow \mathbb{R}$, and also suppose that it's a composition of elementary functions, so its derivative, $T'$, is reasonable easy to find. I can ...
0
votes
2answers
52 views

orthogonal complement problem: show $\operatorname{oc}(A\cap B)=\operatorname{oc}(A)+\operatorname{oc}(B)$

$A$ and $B$ are subspaces of $V$, a finite-dimensional inner product space. Show that $$\operatorname{oc}(A\cap B)=\operatorname{oc}(A)+\operatorname{oc}(B)$$
2
votes
1answer
70 views

Is there a quicker way to show that a set of vectors is a spanning set?

Let's say set $S = \{(2,1,4), (1,-1,1), (3,2,5)\}$ and the vector space V is $R^3$. In this case, the number of independent vectors is equal to the dimension of the vector space. I know that one way ...
1
vote
2answers
51 views

Rank of a Matrix under certain conditions

I am a little confused about the rank of a matrix. When does the rank of a matrix equals to zero? Is rank of a matrix equal to zero when it is a zero matrix or the matrix has no elements in it? Thank ...
1
vote
0answers
54 views

Isometry in Euclidean space

The question is to show that an isometry from $\mathbb{E}^{1} \to \mathbb{E}^{1}$ is of the form $x \to ax + b $ from first principles, and determine the values $a$ can take. From my notes I know for ...
1
vote
0answers
58 views

Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
1
vote
2answers
74 views

fixed point iteration algebra problem

I am looking at an example which finds the root of: $$ f(x) = \cos(3x) \tag 1$$ using the fixed point iteration method. It uses $$ g(x) = \frac{2x+\cos(3x)}{2} \tag 2$$ However, it was my ...
0
votes
1answer
16 views

Norm equality implies parallesim

Let $A$ be some real invertible matrix, and let $u$ and $v$ be two vectors. It is known that $$ ||u ||^2=||Av ||^2 $$ and $$ ||v ||^2=||A^{-1}u ||^2 $$ Does these imply that $u=Av$?
0
votes
3answers
38 views

True or False: If $B=\{b_1,…,b_n \}$ is a base of $R^n$ and for any $1\le i \le n$ exists $v$ so $Av=b_i$ then $A$ is invertible

I have the following homework question: True or False: If $B=\{b_1,...,b_n \}$ is a base of $R^n$ and for any $1\le i \le n$ exists $v$ so $Av=b_i$ then $A$ is invertible I feel this is true but I ...
1
vote
2answers
64 views

A basis for a nilpotent endomorphism [duplicate]

Let $E$ be a complex vector space of dimension 3. Let $f$ be a non zero endomorphism such that $f^2=0$. I want to show that there is a basis $B=\{b_1,b_2,b_3\}$ of $E$ such that $$f(b_1)=0, ...
1
vote
4answers
214 views

Is $(A+B)^2 = A^2 + B^2$ if $A$ and $B$ are matrices

If $A$ and $B$ are matrices is $(A+B)^2 = A^2 + B^2$? I thought because, $AB + BA = AB - AB = 0$.
2
votes
1answer
123 views

Finding the equation of a plane in 3-D by using point-to-point distances

Assume that we have a plane $P(a,b,c,d)$ whose equation is unknown. We know that there is a point set $N = \{n_1, n_2, ...\}$ and $\forall n_i \in N$, $n_i$ is on $P$. Also, $\forall n_i, n_j \in N$, ...
1
vote
2answers
29 views

Proof two solutions of a differential equation are linear independent

Given two solutions for a second order diferential equation: $y(x)=e^{a x}$ and $y(x)=x e^{a x}$ How to show these are linear independent? I procede as follow applying the definition of linear ...
0
votes
1answer
579 views

How to solve an overdetermined linear system given equations with different uncertainties

Please, I would like some help to solve the following problem: I have an overdetemined system of linear equation and want to minimize overall error. Up to now, not a problem, I could use least ...
2
votes
2answers
43 views

Finding an equation of a plane a certain distance from a given plane

I just wanted to know the methodology of how to solve for the equation of a plane that is some distance from some given plane. Thanks. Any help is appreciated
1
vote
1answer
51 views

Dimension of a solution set (Counter example)

I have the felling that the following statement is false: Given an homogeneous system of linear equations expressed as: $$ A \vec{x} = \vec{0} $$ where $A$ has dimensions $n \times m$ with entries ...
0
votes
1answer
20 views

Existence of Rotation-Type Unitary Transformations

I have the following problem If $ x $ and $y$ are points in $n$-dimensional complex space and $|x| = |y|$, then construct unitary matrix $U$ such that $Ux=y$. It seems trivial that the rotation type ...
0
votes
2answers
39 views

Unitary Transformations

This question has stumped me for DAYS... The question: Find a unitary matrix that maps ($1,-1,1$) to ($\sqrt{3},0,0$) and ($1,2,-2$) to ($0,3,0$). What have you tried? I have realized that this ...
1
vote
1answer
26 views

Is $T(P(x))=P'(X)+(x-2)P(X)$ a linear transformation?

Could anyone help me with this question? Let $P_2[\mathbb R],P_3[\mathbb R]$ be the spaces of polinomyals $a_0+a_1x+...+a_nx^n$ where $n \leq 2,n \leq3$ respectively and let $T:P_2[\mathbb R] ...
3
votes
3answers
147 views

If $A \in \mathbb{C}^{m\times n}$ is full-column rank matrix, then is rank($AB$) = rank ($BA$) = rank($B$)?

Let $A \in \mathbb{C}^{m\times n}$, and $B \in \mathbb{C}^{n\times k}$ complex matrices. If A is full-column rank matrix then can we say that rank($AB$) = rank ($BA$) = rank($B$)? What can we say ...
1
vote
1answer
67 views

Hyperplane avoiding some finite set

If R is a K-algebra, where K is an infinite field, does there exist a hyperplane not containing a certain finite set of points? I would appreciate in advance any person answering this question.
0
votes
2answers
45 views

Eigen Value & Eigen Vector Pairwise Relationship

Having same eigen values implies eigen vectors are linearly dependent. But why does it not imply that the eigen vectors are same? Are the eigen value and eigen vector pairs not unique for non-zero ...
2
votes
2answers
774 views

Finding the unknown matrix in an equation?

so I was wondering how can I find the unknown matrix from an equation, I need to find X [-1 2] X [1 0] [-2 -12] [ 0 1] [2 4] = [1 - 4] so I ...
3
votes
3answers
72 views

let $A$ be an $n\times n$ matrix. Show that $\det(A^{-1}) = \frac{1}{\det(A)}$

Let $A$ be a $n \times n$ matrix , and then show that $$\det(A^{-1}) = \frac{1}{\det(A)}.$$ Any tips on this one? basically I don't have a clue.
1
vote
0answers
68 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by producing a linear function

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
1
vote
1answer
22 views

Make a point orbit another point, given time and a normal.

I am working in 3D space. I am trying to make a solar system model. known variables: center of orbit, C (x,y,z) normal, perpendicular to the orbit, N (x,y,z) radius of orbit, R time, position ...
1
vote
1answer
29 views

What cases should I check when I am looking for the possible infinite solutions of a matrix?

I was reading random exercises, and found a typical Determine what values of $a$ cause the system to have no solution, an unique solution and infinite solutions. Also find the solution set for ...
1
vote
1answer
29 views

Volume of parallelpiped question?

I need to find the volume of a parallelpiped. The volume is spanned by 3 vectors $$\begin{cases}a=(-5,-3,2), \\ b=(1,0,2), \\ c=(2,-1,3), \end{cases}$$ so I tried with the formula $(a \times b) \cdot ...
2
votes
1answer
86 views

orthogonal complement of a sum

$A, B$ subspaces of $V$, a finite-dimensional inner product space. SHOW [note oc = orthogonal complement (which is defined below)] 1) $oc(A+B) = oc(A) \cap oc(B)$ 2) $oc(A \cap B) = oc(A) + oc(B)$ ...
1
vote
2answers
59 views

show norm of self-adjoint operator is maximum of abs value of eigenvalue

$M: V \to V$ linear operator show $\|M\| = \max\{|\text{eigenvalue}|\}$
2
votes
2answers
155 views

Is the categorical product for projective spaces essentially the tensor product?

I wonder whether the categorical product of two projective spaces is essentially given by the tensor product of the underlying vector spaces. Is this at least true for projective Hilbert spaces? One ...
1
vote
1answer
34 views

The sum of a normal and nilpotent matrix.

Assume that $n \times n$ matrices $A$ and $B$ are such that $A$ is normal, $B$ is nilpotent, and $A + B = I$. Prove that $A=I$.
1
vote
2answers
71 views

Simple explanation for number of solutions of system of linear equations

So a system of linear equations can be represented as: $$Ax=d$$ where $A$ is a $n\times n$ matrix and $x$ and $d$ are $ n\times 1$ vectors. Now in my notes it says the number of solutions are ...
-1
votes
2answers
23 views

Proving an orthogonal subspace to v.

Let v be a vector in R^n. Prove that the set is a subspace of R^n (called the orthognal subspace to v.
2
votes
1answer
341 views

Inverse of a matrix is expressible as a polynomial?

Let $A$ be an $n \times n$ matrix. Prove that if A is invertible, then there exists a polynomial $p$, such that $A^{-1}=p(A)$ Thus far: Let $W$ denote the $k$ dimensional A-cyclic subspace spanned ...
1
vote
1answer
257 views

differential equations, diagonalizable matrix

I have a question of differential equations of the form. $\textbf{x}'(t)=A*\textbf{x(t)}$, where x is an n-dimensional matrix, and A is an n*n real matrix. I have learned to solve this if a is ...
0
votes
1answer
84 views

What is bi-infinite matrix?

I am reading a wavelet analysis book. In one section there is a term "bi-infinite matrix". I have searched a lot but has not found a good definition. So can anyone tell me, in concise, what's the ...
4
votes
1answer
92 views

A Question about Non-Conservative Vector Fields

In my multivariable calculus class, we spent some time discussing the vector field that was the gradient of arctan(y/x). This field was shown to be non-conservative in closed regions which enclosed ...
1
vote
1answer
22 views

T-invariance #homework

Let $V$ be a vector space and let $T \in L(V)$. Given a $T$-invariant subspace $U$ is it true that exists a $T$-invariant space $W$ such that $V = U \oplus W $?
0
votes
2answers
49 views

Help finding an eigen vector

Find the eigen vectors of I found $v=\left[1,2/3,1\right]$ and $\left[-1,0,1\right]$ but according to wolfram, there is one more, $\left[0,1,0\right]$, does ...
1
vote
2answers
58 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
0
votes
1answer
24 views

Help with finding eigenvectors

Find eigen vectors for this: I found that eigenvalues are $0,2,2$ And the eigen vector for $0$ is {$1,0,1$} But I'm not sure how to find the eigenvector for ...
0
votes
1answer
90 views

Please, help me locate the steps in this induction problem

I am having hard time locating $p(n)$, $p(1)$, $p(n + 1)$, $p(n) \to p(n + 1)$ in the proof below. Please, help me find the borders of induction steps. Thanks. Let $(**)$ $$ \begin{matrix} ...
0
votes
2answers
34 views

Proving that an eigenvalue is a root of a polynomial

Let $A$ be an $n \times n$ matrix, and let $\lambda$ be an eigenvalue of A. Prove that if $p$ is a polynomial such that $p(A)=\mathbb{0}$ then $\lambda$ is a root of $p$.
3
votes
3answers
404 views

If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$

How can you show that if $B$ is a maximal linearly independent set of $V$, then this implies that $B$ is a basis of $V$?