Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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finding a power of a matrix

When you are given the eigenvectors and eigenvalues of a matrix A and are asked to solve for A^3, for the formula A = PDP^-1 is your diagonal matrix the identity matrix with the eigenvalues swapped ...
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29 views

How to determine column dependency without calculating the determinant?

Determine whether this matrix' columns are linearly dependent or not. $$\begin{bmatrix} 1 & 0 & 2 \\ 0 & -1 & -2 \\ 2 & -2 & 0 \end{bmatrix}$$ The determinant is $0$ ...
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27 views

Orthogonal projection onto a plane

Find the minimal distance from the point $P = \begin{bmatrix}\\ -8 \\ 14 \\ 8 \end{bmatrix}$ to the plane $V$ of $\mathcal{R}^3$ spanned by $\begin{bmatrix}\\ 1 \\ 2 \\ -2 \end{bmatrix}$ and ...
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55 views

Sesquilinear forms seen as bilinear maps

Let $V$ be a complex vector space. A sesquilinear map (or conjugate-linear in the first variable and linear in the second) on a complex vector space $V$ is a map $f: V \times V \rightarrow \mathbb{C}$ ...
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Besides being symmetric, when will a matrix have ONLY real eigenvalues?

I realize that when a matrix is symmetric, then it must have all real eigenvalues. However, I am doing research on matrices for my own pleasure and I cannot find a mathematical proof or explanation ...
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Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
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23 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
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82 views

How to calculate a linear combination for a matrix' column?

I have a very weak understanding of linear dependency and linear combination, so I figured I'd check out some exercise about it: $$A = \begin{bmatrix} 4 & 0 & 1\\ 2 & 3 & 6\\ 6 ...
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54 views

Explain why each set is NOT a basis for the given vector space

My biggest problem with linear algebra is trying to get the wording right when I answer questions. I want to communicate my answers as effectively as possible. So here are my answers to the following ...
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33 views

three dimensional subspace question

If a vector is in $\mathbb{R}^5$, does this mean that the projection of this vector onto $S$ is in $\mathbb{R}^3$, where $S$ is some 3-dim subspace of $\mathbb{R}^5$?
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Going from $X^tAB - I = X^t$ to $X^t(AB-I) = I$ in matrix algebra.

Finding the value of the matrix $X$: $$X^tAB - I = X^t$$ I noticed that the next step chosen by my book is $$X^t(AB-I) = I$$ It's not clear to me how did they reach that. How did they go from one ...
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Algebraic and Geometric Multiplicity

I am having a hard time understand these two concepts Algebraic multiplicity and Geometric multiplicity of a matrix regarding its eigenvalues for example if I have the matrix: ...
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41 views

Proving vector projection

For nonzero vectors, how do you prove the following? $$\|u\|^2 = \|\text{projection of $u$ onto $v$}\|^2 + \|u - \text{projection of $u$ onto $v$}\|^2$$ I think what we need to do is split up the ...
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23 views

I need help understanding what r-th and s-th rows are.

Let E be the matrix obtained from the unit $n \times n$ matrix by multiplying the $r$-th row with a number $c$ and adding it to the $s$-th row, $r \neq s$. Let $A$ be an $n \neq n$ matrix. Then ...
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28 views

Calculating matrix determinants based on another's.

$$A = \begin {bmatrix} a & b & c \\ 4 & 0 & 2 \\ 1 & 1 & 1 \end {bmatrix} \ \ , \ \ \left| \ A \ \right| = 3$$ Knowing only this, how does someone calculate the determinant ...
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103 views

$3 \times 3$ matrix with entries $-1$ and $1$

There are $512$ matrix due to $2^9$. Is there a way instead of by hand to find how many of the matrix may equal $1, 2, 3....,$ etc. with the entries being $1$ and $-1$? Thanks in adavance.
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96 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 ...
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101 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 ...
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24 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?
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114 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 ...
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42 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 ...
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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 ...
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44 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 $$\operatorname{oc}(A\cap B)=\operatorname{oc}(A)+\operatorname{oc}(B)$$
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51 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 ...
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39 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 ...
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42 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 ...
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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 ...
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61 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 ...
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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 $u=Av$?
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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 ...
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48 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, ...
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198 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, $AB + BA = AB - AB = 0$.
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120 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, $\forall n_i, n_j \in N$, ...
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28 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 ...
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131 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 ...
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42 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
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31 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 ...
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1answer
19 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 ...
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35 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 ...
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23 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] ...
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93 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 ...
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58 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.
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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 ...
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Hyperplane - soft and hard questions

This would be a rather long question. Apologies for that. Instead of asking three separate questions I've consolidated them in one. I am trying to learn hyperplanes, convex hulls, separations theorems ...
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495 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 ...
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3answers
69 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.
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63 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 ...
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1answer
19 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 ...
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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 ...
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25 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 ...