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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How to find the restrictions of side length on an obtuse triangle

Question: In Triangle ABC, the angle ∠ABC is an obtuse angle. The Side AB is 1cm, and the side BC is 3cm. Side AC is (3x+10)/(x+3) cm Find the restriction(s) on x. I have tried a few different ...
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1answer
38 views

Solve $a-1.73d=0, b-1.73d=0, c-1.73d=0, a+b+c -1.73d=0$ [closed]

How can we find nontrivial solutions of the homogeneous equation $$a-1.73d=0, b-1.73d=0, c-1.73d=0, a+b+c -1.73d=0$$ I need to find the values of $a,b,c$ and $d$. When I tried with Gauss ...
2
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1answer
25 views

Two real symmetric matrices are congruent if and only if they have the same rank and signature.

So I saw this statement in an exercise : Two real $n \times n$ matrices are congruent if and only if they have the same rank and the same signature. But I was wondering why do we need to state ...
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3answers
32 views

Properties of RREF 3x3 matrix is the identity

The row reduced echelon form of a 3 × 3 matrix A is the identity. State whether each of the following is true or false. You do not need to explain your answers. (a) A has an inverse. (b) The columns ...
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38 views

What is the subspace of the particular solution to $Ax = b$?

If I solve the equation $$ \begin{bmatrix}1 & 2 & 1 \\ 3 & 2 & -1 \\ -1 & 2 & 3\end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} = \begin{bmatrix}0 \\ 4 \\ -4\end{...
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2answers
38 views

Computing the span

Compute the span of $\begin{bmatrix} 4\\2\\10 \end{bmatrix}$ and $\begin{bmatrix} 6\\3\\15 \end{bmatrix}$ I just don't even understand what "Compute the span" is even asking me. Can anyone give me a ...
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2answers
41 views

show that Row(A) is perpendicular to Null(A)

Hi can you please help me check my work Question: Prove that if A is a m x n matrix, vector x is an element of Row(A) and vector y is an element of Null(A), then vector x is perpendicular to y. ...
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3answers
53 views

How do you compute the span of a 3x1 matrix?

How do you compute the span of a 3x1 matrix? for Example: Compute the span of $\begin{bmatrix} 4\\0\\1 \end{bmatrix}$ and $\begin{bmatrix} 1\\0\\4 \end{bmatrix}$
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2answers
31 views

Write the vectors u, v, w, z in terms of a and b.

Write the vectors $u, v, w, z$ in terms of $a$ and $b$. I'm unsure of how to do this.. If someone could give me an example of one being done I'm almost positive I could mimic it and figure out the ...
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0answers
41 views

Incorrect answer - Simultaneous Differential Equations

The questions states solve for y such that $$y' = \begin{bmatrix} -4 & 2 & 1 \\ 1 & -3 & 1 \\ 3 & -3 & -2 \\ \end{bmatrix}y , y(0)= c = \begin{bmatrix} 1\\5\\3 \end{...
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1answer
45 views

Prove set of 5 elements of $M_{2\times 2}$ is linear independent

I want to prove if a set of $5$ elements of $M_{2\times 2}$ is linearly independent. Since I have $5$ elements I think it's impossible for it to be independent since when I sum up these matrices I'll ...
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1answer
59 views

what are the possible values of dim(Null(L))?

can you please explain this question to me? Thanks Question : Suppose that L: R^4 ----> R^2 is a linear transformation. a) what are the possible values of dim(Null(L))? b) For each possible value ...
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1answer
32 views

Help with proving a 2 by 2 determinant is the area of parallelogram

I have proved a large part of this by the following but get stuck at the last step. To say $A=ad-bc$, we still need $ad>bc$. I have puzzling over this for hours. Thank you!
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2answers
28 views

Proving $(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$ , $P$ idempotent matrix.

Given that a matrix $P$ is idempotent how to prove the following relation: $$(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$$ $c$ is any real constant.
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1answer
24 views

Collinear Points in 3-Dimensions

The points A(3, -1, z), B(1, 2, 6), and C(x, 8, 14) are collinear. Find the values of x and z. I have tried finding common ratios between the points, but no common ratio is possible, I have a feeling ...
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3answers
52 views

Method of Proof in Showing Something is Smallest (Subspace)

I am reading a proof that shows the sum of subspaces is the smallest subpsace containing all the summands (It is a vector space over $\mathbb{R^n}$). The author of the book goes to show first it is a ...
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1answer
28 views

Power method and convergence

I am working on some practice problems for the convergence of power method for some given recursion relationship and I am trying to generalize/reflect on the question after having been stuck on the ...
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1answer
31 views

Transformation matrices in a basis

Let $F=\mathbb{R}$,$X=\mathbb{R^3}$ and $Y=\mathbb{R^2}$. Further $B_X$ and $B_Y$ are given by: $B_X:=\{(1,0,0),(1,0,-1),(1,-1,-1)\}$ $B_Y:=\{(1,0),(1,-1)\}$ Let $f:X\rightarrow X$ and [...] be ...
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2answers
18 views

Prove if set of P4 is linearly independent

Hi I'm trying to prove if this set is linearly independent $s=1-x^4,1+x+x^3+x^4,1+x-x^3+x^4,1-x^2$ I need to prove $c1(1-x^4)+c2(1+x+x^3+x^4)+c3(1+x-x^3+x^4)+c4(1-x^2)=0$ So I've rearranged the ...
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2answers
21 views

Dimension of the span of two parallel lines in $R^4$.

I am asked if the following question is true or false: Let $r,s$ be two parallel lines in $R^4$ then the dimension of $Span(r \cup s)$ is strictly less than $3$. I think this is true because two ...
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1answer
47 views

Computing the standard matrix of the linear transformation

Can you please explain this question to me? Suppose that $w = [1,2,3]^T$ and $L: \mathbb{R}^3\to \mathbb{R}^3$ is defined by $L(x) =\text{Proj}_w(x)$ (projection of $x$ onto $w$). Compute the ...
2
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1answer
39 views

Parametrized linear operator

I've been trying to solve the following task: Determine $a$, $b$ $\in \mathbb{R}$ so that for the linear mapping $A :\mathbb{R}^3\to\mathbb{R}^3 $, with linear transformation matrix $$\mathcal{M}(...
2
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1answer
44 views

What exactly does a rotation preserve?

I understand a rotation should preserve length and angle and hence the dot product. Since anything that preserves the dot product is a linear transformation, then a rotation can be represented by a ...
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1answer
35 views

What is the quotient space $\mathbb{R}^2/F$ where $F =a[1,1]$?

What is the quotient space $\mathbb{R}^2/F$ where $F =a[1,1]$? How do you find a basis of it? What is a good way to think of it in mind?
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1answer
19 views

Find the value of n , using eigenvector

I am unable to think how shall I proceed. I have to find value of n given a 2×2 matrix and an eigenvector. Can somebody help me out.
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3answers
30 views

How many convexly independent vectors there are in $\mathbb{R}^n$

I know there are n linearly independent and n + 1 affinely independent vectors in $\mathbb{R}^n$. But how many convexly independent there are? I think there are infinity number of them because if I ...
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0answers
26 views

orthogonal basis and complex number

Let $v_1 =(1,i,2+i)$, find vectors $v_2$ and $v_3$ so that $α=${$v_1,v_2,v_3$} is an orthogonal basis for $C^3$. i understand how to find orthogonal basis in $R^n$ through gram schmidt process but ...
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2answers
55 views

For an orthogonal matrix $Q$, why does $QQ^T = I$?

In my linear algebra text (Strang), an orthogonal matrix is defined to be a square matrix whose columns are orthonormal. In other words, an orthogonal matrix is a matrix $Q = [q_1 \cdots q_n]$ where ...
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4answers
49 views

A question on linearly independent vectors in a Banach space

Given a list of linearly independent vectors $\{x_1,...,x_n\}$ in a Banach space. If for each $1 \leq i\leq n$, there is a sequence of vectors $\{y_m^{(i)}\}_{m=1}^{\infty}$ converges to $x_i$. Then ...
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0answers
21 views

transformations and change of basis

I was wondering if someone could help to explain the notation of this form as well a possible example of questions that involve such notation (is diagonalizing a matrix etc. an example?).
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2answers
52 views

Check if a positive solution exist of a linear equation with two variables?

Let's say there's an equation $$a x + b y = c$$ where $a,b,c > 0$ are given. I want to know if positive solutions $x, y >0$ exist for this equation.
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1answer
35 views

Algebra problem about Ker and Im

I have a problem with this linear algebra exercise. A) Find an orthonormal basis with respect upon the Euclidean product for a vector space in $\mathbb{R}^3$, generated by those vectors: $(1, 2, -1)$...
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2answers
31 views

matrix calculation

Let $p= \begin{pmatrix} x & y \\ z & v \end{pmatrix}\in M_2(\mathbb{C})$ such that $p^2=\overline{p}^t=p$ and rank(p)=1. Why is $p=\begin{pmatrix} t & l\sqrt{t(1-t)} \\ \overline{l}\sqrt{...
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1answer
33 views

On the dimension of subspaces of the vector space given by the product of polynomials.

I was asked this question orally so feel free to also correct how the question is written. Given the vector space of polynomials in the variable $x$ with degree $\le 4$ and the vector space of ...
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1answer
28 views

Exact same solutions implies same row-reduced echelon form?

In Hoffman and Kunze they have two exercises where they ask to show that if two homogeneous linear systems have the exact same solutions then they have the same row-reduced echelon form. They first ...
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1answer
35 views

Upper and lower bounds log determinant

I found an inequality in Wikipedia that i want to know how to prove it. For a positive definite matrix A, the trace operator gives the following tight lower and upper bounds on the log determinant. $...
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1answer
36 views

How to prove that $A$ is positive semi-definite if the symmetric minors are non-negative?

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix such that all its symmetric minors are non-negative (i.e. for $B=\left(a_{l_il_j}\right)_{1≤i,j≤k}$ with $1≤l_1<...<l_k≤n$ we have $\det(B)≥...
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0answers
38 views

At which points is function invertible?

Determine on which points is mapping local invertible? $f: \mathbb R ^{2}\mapsto \mathbb R ^{2}$ $\left(x,y\right)\mapsto\left(x^{2}-4y^{2}+x, -xy+3y \right) $ I calculated Jacobian matrix and ...
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1answer
27 views

Isomorphism of vector spaces - is 1:1 enough?

From 'Functional Anaylsis' by Bachman If the linear transformation is 1:1, it is called an isomorphism Is this right? I thought an isomorphism was a morhpism that admitted an inverse. Unless we ...
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1answer
44 views

Show that function $\mathcal F$ is norm preserving

$N \in \Bbb N$. The function $\mathcal F $ in $(\Bbb C ^N , || \cdot || _2 )$ is defined as follows: $$ (\mathcal F (x))_k := \frac 1 {\sqrt N} \sum^N_{j=1} x_j \mathrm {exp} ({2\pi i \frac {(j-1)(k-1)...
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Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
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0answers
25 views

Isomorphism and linear transormation

Is there Isomorphism $T:R_{3}[x]\to R^{3}$: $T(x^2+2x)=(1,2,1),T(x+1)=(0,1,1),T(x^2-2)=(1,0,-1)$ I know that first of all, I need to prove that $T$ is a linear transformation by making a base, but I ...
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1answer
38 views

Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
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2answers
102 views

$A^{n+1}=0\Rightarrow A^n=0$

A real $n\times n$-matrix $A$ satisfying $A^{n+1}=0$ must necessarily satisfy $A^n=0$. One way to see this is by looking at the Jordan Normal Form of $A$, another is by an argument involving the ...
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1answer
28 views

existence of inner product preserving linear map?

I want to prove this: Given a vector space $V$ on $\mathbb{R}$ with a positive definite inner product $\left \langle .,. \right \rangle$. Show that there exist a natural number $p$ and a linear map $...
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1answer
43 views

Geometric meaning of Equation

As a part of my linear-algebra exam preparation, I am going through the surface equation and quadratic-bilinear form usage in my book which is a part we haven't really went through and left to explore ...
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3answers
45 views

Orthogonal complement and projection

Let $M$ be a subspace of $\mathbb R^4$ which is spanned by the vectors $v_1 = (1,0,-1,1)$ , $v_2=(0,1,2,1)$. Find the orthogonal complement $M^T$ of $M$ and the orthogonal projections of the vector $v=...
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21 views

Can I do Gaussian elimination with a rectangular matrix

I'm writing some linear algebra scripts to understand the stuff I'm reading. I was wondering if I can only use square matrices as input for Gaussian Elimination, my guess is yes because permutation ...
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1answer
15 views

3-Space Vertices of a Parallelogram

The points (1, -2, 4), (3, 5, 7) and (4, 6, 8) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vertex, and find the three ...
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1answer
30 views

Recursion relationship and linear algebra

I wanted to confirm my intuition about a problem that I got wrong relating to an application of the power method to recursion relations. The question is as follows: For the context of the question, ...