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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Conversion from Cartesian to Parametric function for a plane

I am given a plane in $\mathbb{R}^3$ with Cartesian equation $$ -5 x_1 - 2 x_2 + 2 x_3 = -15 $$ and I would like to find parametric equations $$ \mathbf{x} = \mathbf{x}_0 + t_1 \mathbf{v}_1 + t_2 ...
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Basis for the null space of an identity matrix

Is the set containing only the zero vector a basis for the null space of an identity matrix?
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60 views

Proof that $(A+B)^T=A^T+B^T$ (homework question)

Homework question: Proof that $(A+B)^T=A^T+B^T$ Let A and B be $m \times n$ matrices. Prove that $(A+B)^T=A^T+B^T$ by comparing the ij-th entries of the matrices on each side of this equation. (Let ...
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80 views

Linear dependence when number of vectors is greater than/less than the dimensions of the vector space

Simple question here, I just need some clarification of a theorem. Theorem: if k > n, then any k vectors in $R^n$ are linearly dependent. Nice and easy I guess! My question is this: Does this imply ...
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135 views

Eigenvalues of checkerboard matrix

I am trying to find the eigenvalues and eigenvectors of the following 4x4 "checkerboard" matrix: $$ \mathbf C = \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 ...
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91 views

Proving the equations $x_1+\dots+x_n=0$, …, $x_1^n+\dots+x_n^n=0$ have a unique solution

Let equations of the form $\left\{\begin{matrix} x_{1}+x_{2}+...+x_{n}=0\\ x^{2}_{1}+x^{2}_{2}+...+x^{2}_{n}=0\\ .........\\ x^{n}_{1}+x^{n}_{2}+...+x^{n}_{n}=0 \end{matrix}\right.$. Proof: ...
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2answers
77 views

$A$ and $B$ are similar. why $rank(A)=rank(B)$?

$A$ and $B$ are similar matrix in ${c^n}$. why $rank(A)=rank(B)$?
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39 views

A,B are diagonalizable matrix and their characteristic polynomials are the same.prove that $A$ and $B$ are similar

let A,B are diagonalizable matrix in ${c^n}$ and their characteristic polynomials are the same. can we prove that $A$ and $B$ are similar?
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46 views

A is the matrix of $T$: $V \rightarrow V$ with repect to the basis H. Find the matrix B of $T$ with respect to G.

A is the matrix of $T$: $V \rightarrow V$ with repect to the basis H. Find the matrix B of $T$ with respect to G. $A = \begin{pmatrix} 1 & 2 & 1\\ 0 & 1 & 1\\ 1 & 0 & 2 ...
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117 views

What is the real meaning of the determinant of a matrix? [duplicate]

I understand how it is calculated, but what does the determinant of a matrix mean and what is it used for? If you could explain it to me in simple words, thank you.
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79 views

Can you multiply a matrix with a determinant without expanding the determinant?

Say it is a 3x3 matrix. Can you multiply it with a 3x3 determinant without expanding and calculating the determinant's value? Thank you.
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51 views

determine if the given function is a linear transformation.

Determine if the given function is a linear transformation.$T : R^n \rightarrow R^n$ with T(v)= $[0 ... 0]$ My thought process on solving this: I know that in order to be a linear transformation, ...
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1answer
63 views

Linear algebra minimal sum

Among all the unit vectors $u=(x,y,z)$ in $\mathbb{R}^3$, find the one for which the sum $x + 2y + 3z$ is minimal. How do I get the minimal? I know the unit vectors of $\mathbb{R}^3$ but do I ...
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36 views

Matrix columns and independence

So I'm studying for an exam and solving this problem. I've been watching countless online tutorials and reading books but I'm still not 100% if I'm doing this correctly since there's many different ...
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221 views

Question regarding arbitrary parameters

Solve the following system of linear equations: x + y + z = 4 x + y + z = 4 2x + 2y + 2z = 8 I'd like some help understanding how to go about solving this. I ...
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1answer
60 views

Proof of UNIQUE SOLUTION- to $x^TAy=f^{T}y$ for any $y\in \mathbb{R^n}$

Here is the original problem: **Define the bilinear bounded and elliptic map: $B:\mathbb{R^n}\times \mathbb{R^n}\to \mathbb{R}$ as follows: $B(x,y)=x^{T}Ay$ where $A$ is an $n\times n$ matrix. ...
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28 views

Find the general formula for $x_k.$

Suppose that the sequence $x_0, x_1, x_2...$ is defined by $x_0 = 0, x_1 = 1, x_2 = 5,$ and $$ x_{k+3} = 8x_{k+2}−17x_{k+1}+10x_k$$ for $k≥0.$ Find a general formula for $x_k$. Putting in the format ...
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217 views

What makes the Cauchy-Schwarz inequality so important?

The Cauchy-Schwarz inequality is $(a\cdot b)^2 \leq |a|^2|b|^2$. Why is this considered such an important inequality: to quote my textbook it's "one of the most important inequalities in all of ...
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0answers
44 views

Finding a function using first derivative

I have some data about just first derivative of a function. Also, I know a point of this function(e.g. (x1,y1)). How can I obtain the function? All my date are numerical. dev f(x)=[ 580.00 , 479.7308 ...
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38 views

How do I convert this to a linear programming problem?

It takes a tailoring 2 hours of cutting and 4 hours of sewing to make a knit suit. To make a worsted suit, it takes 4 hours of cutting and 2 hours of sewing. At most 20 hours per day are available for ...
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87 views

How do I convert this into a linear programming problem?

A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of \$50 and each acre of barley yields a profit of \$70. To sow the crop, two machines, a tractor and tiller, are ...
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42 views

Operator matrix norm associated with the vector norm

What is the operator matrix norm associated with this vector norm? $$\Vert x \Vert = \frac{1}{n} \mathop {\sum} \limits_{j=1}^{n} \vert x_j \vert, \qquad (x \in \mathbb{R^n})$$
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58 views

Is it possible that $\Vert I \Vert > 1$ !?

For any matrix norm, is it possible $\Vert I \Vert > 1$ ?, where $I_{n\times n}$ is identity matrix. If not, why in some books they write $\Vert I \Vert \geqslant 1 $ ?
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Spectral Theorem / Quadratic Form Minimization Problem

Here is the problem: Let $A$ be an $n \times n$ symmetric matrix. Let $S = \{ \mathbf x \in \mathbb R^n : ||\mathbf x|| = 1 \} $ denote the unit sphere. Let $Q(\mathbf x) = \mathbf x ^TA\mathbf x $ ...
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93 views

Show that every $A \in SL_3-$action has at least 3 fixed points on $\mathbb{P}^2$.

Consider the natural action of $SL_3(\mathbb{C})$ on $\mathbb{P}^2$ via: $$ SL_3(\mathbb{C}) \times \mathbb{P}^2 \to \mathbb{P}^2, \ (A,[v]\mapsto [Av]). $$ It is clear that the kernel of the ...
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59 views

Modelling Problem in Linear Programming Standard Form

I'm having a hard time setting this up, so that's what I need help with. The solving I understand. We’re making a drink with the following requirements: at least 500 calories, at least 20 mg. of ...
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1answer
57 views

What are all angle preserving linear operators on $\mathbb R^n$?

I´m working on Spivak's Calculus on Manifolds and I met this exercise. My immediate answer was 'all the rotations' but I can't explain why. Am I right? Can you give a hint or something to be able to ...
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1answer
31 views

Linear Transformations

I have no idea how to work this question. Can someone offer some insight?
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2answers
56 views

How to solve this eigenvector eigenvalue problem?

Let $L: P_1 \to P_1$ be the linear operator defined by $L(at+b)=-bt-a$. Find, if possible, a basis for $P_1$ with respect to which $L$ is represented by a diagonal matrix. How about this?
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Jordan form, similarity and eigenvalues

A complex matrix is always diagonalizable in the Jordan canonical form (right? ). also, two matrices have the same Jordan form if and only if they are similar. But two matrices who have the same ...
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2answers
67 views

Induction question with 'if' statement

I have an induction homework question that I got stuck in the middle. Prove by induction that if $a + a^{-1} \in \Bbb{Z}$ then for each $n \in \Bbb{N}$ the following is true: $$a^{n} + a^{-n} \in ...
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1answer
37 views

proof to proposition involving dimension

i came across the dimension theorem as a proposition in linear algebra. dim(w1 + w2)=dim(w1)+dim(w2)+dim(w1 n w2). w1 and w2 are subspaces of the vector space V. I am trying to prove this using bases ...
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Consider the map defined by $T:\mathbb R^3 \to \mathbb R^2$ defined by $T(x_1 -2x_2 + x_3, 2x_1-3x_2 + x_3)$.

I'm asked to find $N(T), R(T)$, a basis for them, and to determine whether it is onto or 1:1. I wrote my mapping in matrix form and I get $$\begin{matrix} 1 & -2 & 1 \\ 2 & -3 & 1 \\ ...
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89 views

A certain kind of rank $2$ matrices can be written as a sum of two rank $1$ matrices of the same kind.

I want to prove that a rank $2$ matrix of the form $$ \left( \begin{array}{ccc} X_{0} & X_{1} & X_{2} \\ X_{1} & X_{2} & X_{3} \\ X_{2} & X_{3} & X_{4}\end{array} ...
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1answer
206 views

Hoffman “Linear Algebra”: why need such a long proof?

I'm reading "Linear Algebra" by Kenneth Hoffman and Ray Kunze. I don't quite understand why there's a long proof in $\S$6.4 Theorem 6. First the triangular matrix is defined: An $n\times n$ ...
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2answers
75 views

Definition of orthogonal projection

The concept of the orthogonal projection is an easy one to grasp, but I'm confused about the following definition in my book: Let $\{u_1,\dots,u_k\}$ be an orthonormal basis of a ...
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1answer
48 views

Proof approach: A 7x7 matrix with 15 ones can allow at least three marriages

This is quite difficult to prove imho with regards to Hall's Marriage Algorithm I can visualize a number of scenarios that work (i.e. put ones from the first entry to the fifteenth, or across ...
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102 views

Orthogonal complement of a complex subspace

Let $W$ = Span{$(1,i,0), (0,1,-1)$} Find $W^\perp$ If $W$ were not a complex subspace, I would solve this by finding the kernel of the rowspace of the corresponding matrix. However, if I do this on ...
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Find a finite set of vectors which spans $W$.

Let $W$ be the set of all $(x_1, x_2, x_3, x_4, x_5)$ in $\Bbb R^5$ which satisfy $2x_1-x_2+{4 \over 3}x_3 - x_4\qquad = 0$, $x_1\qquad+{2 \over 3}x_3\qquad- x_5 = 0$, $9x_1-3x_2+6x_3-3x_4-3x_5 = 0$. ...
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1answer
30 views

Prove $V_{e} + V_{o} = V$

Prove $V_{e} + V_{o} = V$ where $V_{e}$ is a subset of even functions from $R$ into $R$, $V_{o}$ is a subset of odd functions from $R$ into $R$. I have proved $V_{e}$, $V_{o}$ are subspaces and ...
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26 views

Prove that this linear operator on the space $\mathbb{R}$[x]$_n$ has the set of eigenvalues $1,a,…,a^n$

I'm trying to show that the linear operator $f$ $\mapsto$ $f(ax+b)$ on the space $\mathbb{R}$[x]$_n$ has the set of eigenvalues $1,a,...,a^n$ I started this question like this $Tf = \lambda$$f$ ...
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32 views

Consequence of linear combination in matrix .

If a column of a matrix is linear combination of another column, what are the consequences ? Several terminology coming into my mind to relate with this such as Rank of the matrix ; Determinant ...
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1answer
25 views

Show that the vectors form a basis for $R^3$.

Show that the vectors $\alpha_1 = (1, 0, 1)$, $\alpha_2 = (1, 2, 1)$, $\alpha_3 = (0, -3, 2)$ form a basis for $R^3$. Is it enough to show that the vectors are linearly independent?
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Invertible and non-invertible linear transformation

If a linear transformation is represented by an non-invertible matrix $P$, then it might happen that two different vectors (points in $\mathbb{R^n}$) will be mapped to the same point. However, if the ...
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Can we use $R_1 \to R_2-R_1$ as an elementary row operation without changing the value of the determinant?

Can someone help me out? I need to know if $R_1 \to R_2-R_1$ is a valid elementary row operation that can be used on the given determinant without changing the determinant's value. It's a $3\times 3$ ...
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80 views

Can we interchange one row and one column in a determinant?

Can we swap the ith row and the ith coloumn in a determinant as an elementary operation? What happens when we do interchange them? Does the value of the determinant remain constant or does it change?
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59 views

Showing that Determinant is a Volume Multiplier

I want to show using the change of change of variables theorem for (Riemann) integration that the determinant of a linear transformation $T$ is a scaling factor for the volume of a space. If $1_A$ is ...
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1answer
119 views

Help with a tricky matrix equation

Say I have the following variable length vectors containing unknown values: $$ A=\left (\begin{array}{c} a_1 \\ a_2 \\ \vdots\\ a_i\\ \end{array}\right) B=\left (\begin{array}{c} b_1 \\ b_2 \\ ...
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1answer
62 views

Complex numbers - addition of two modulus help

Ok, so I got the answer to part i), but however, I'm not so sure how to get the answer to part ii). The answers say its an ellipse and they specified the equation, but I can't understand how they came ...
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69 views

Gram-Schmidt Process and Orthogonal Components

Let the Gram-Schmidt process transform the vector system $(a_{1}, ..., a_{n})$ into the system $(b_{1}, ..., b_{n})$. Show that the vector $b_{k}$ is the orthogonal component of the vector $a_{k}$ ...