For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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1answer
35 views

Is my proof about minimal polynomial correct

I am to prove that the characteristic polynomial and minimal polynomial have same roots. That is, if $\lambda$ is an eigenvalue of the linear transformation $T$ and if $p(t)$ is the minimal polynomial ...
-3
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2answers
43 views

linearly dependent or linearly independent [closed]

Is it true that if $\vec{u_1}$ , $\vec{u_2}$ and $\vec{u_3}$ are linearly dependent then $\vec{v_1}$ = $\vec{u_2}$ + $\vec{u_3}$ , $\vec{v_2}$ = $\vec{u_1}$ + $\vec{u_3}$ , $\vec{v_3}$ = $\vec{u_1}$ + ...
1
vote
1answer
49 views

Closest Vector in a Inner Product Space

Let $V$ = $\mathbb{R}^n$ Note that $\langle -,-\rangle$ defines the Inner Product on $\mathbb{R}^n$ $$\|v\| = \sqrt{\langle v,v \rangle}$$ Consider the standard Distance Function $$d(x,y) = ...
1
vote
1answer
31 views

Clarification between a module and a vector space?

I'm reading Kenneth Hoffman's Linear Algebra, Ed2. In $\S5.5$ it talks about Module and Vector Spaces: (1) If $K$ is a commutative ring with identity, a module over $K$ ( or a $K$-module) is ...
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0answers
20 views

Do affine spaces have coordinate transformations?

I asked a question on Physics SE and there seemed to be some confusion as to whether affine spaces could have coordinate transformations. Specifically, the particular space I was working with was ...
1
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2answers
33 views

What would be a characterization of a definite operator?

Let $V$ be an $n$-dimensional inner product space and let's call $T\in \mathcal L (V)$ definite if $$\forall x \neq0: \langle Tx,x\rangle \neq 0. $$ An obvious sufficient condition for $T$ to be ...
1
vote
1answer
41 views

Vector Space Subspace Proof

Suppose that W is a subspace of a finite-dimensional vector space V . Prove that W = V if and only if dim W = dim V. This is what I did: Suppose dim W =dim V$\iff|$basis of $W|=|$basis of $V|$ and ...
3
votes
1answer
33 views

Equation of a plane containing a straight line

A straight line passes through the points $(1, 2, 3)$ and $(-3, -2, -1)$. I have calculated the system of equations of this line to be $$ x = 1 - t,\, y = 2 - t, \, z = 3 - t $$ The question I ...
1
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1answer
34 views

Argument that Vandermonde matrix's determinant has $n-1$ distinct roots

det(Vandermonde) = $\left|\begin{array}{ccccc}1 & x & x^2 & ... & x^{n-1} \\1 & a_2 & a^{2}_{2} & ... & a^{n-1}_{2} \\1 & ... & ... & ... & ... \\1 ...
0
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1answer
20 views

To prove neccesary and sufficient condition for set W to be a subspace

The necessary and sufficient condition for a non - empty subset W of a vector space V(F) to be a subspace of V is $a$,$b$ in F and $\alpha$ , $\beta$ in W implies a$\alpha$ + b$\beta$ in W I need ...
1
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0answers
24 views

Orthogonal spaces, span-formula

I read in a book this formula: ($v_i$ are vectors of an euclidean vector space, each one $\neq$ 0) $(\cap v_i ^\bot )^\bot = \sum v_i^{\bot \bot}$, The intersection and the sum are build over a ...
10
votes
1answer
198 views

Does the vector space of compactly-supported continuous functions $X \rightarrow \mathbb{R}$ satisfy an interesting universal property?

Let $S$ denote a set. Then the vector space $FS$ freely generated by $S$ can be identified with the set of all finitely-supported functions $S \rightarrow \mathbb{R}$. This gave me the following idea; ...
2
votes
3answers
144 views

Find Transformation Matrix $T$ relative to new bases such that $T$ is in diagonal form

$T$ is a linear transformation from $R^2 \rightarrow R^3$. The matrix of $T$ = $\left[\begin{array}{cc}1 & -1 \\0 & 0 \\1 & 1\end{array}\right]$. Question: how to find bases $(e_1,e_2)$ in ...
1
vote
1answer
16 views

For which values of $a_{ij}$, $<u,v> = a_{11}x_1y_1+a_{12}x_1y_2+a_{21}x_2y_1+a_{22}x_2y_2$ is an inner producy

Let $u = (x_1,x_2)$ and $v=(y_1,y_2)$ Then, when: $<u,v> = a_{11}x_1y_1+a_{12}x_1y_2+a_{21}x_2y_1+a_{22}x_2y_2$ is an inner product? The exercise asks me first to prove that the additive and ...
0
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1answer
25 views

If $T$ is an isomorphism and $<u,v>$ an inner product, then $<T(u), T(v)>$ is also a inner product.

I have to prove that if $T$ is na isomorphism, then $<T(u), T(v)>$ is also an inner product, when $<u,v>$ is an inner product. I've tried to assume that, since $<u,v>$ is an inner ...
0
votes
0answers
43 views

Why do sines and cosines form a basis, and can be considered a vector space?

Many times I've seen that Fourier series are justified because we are thinking that the set of all functions of the form $sin(ax)$ and $cos(ax)$ form a vector space. A function can therefore be ...
0
votes
1answer
19 views

Vector space spanning proof.

Question: Consider in a real vector space $V$ the subspace $U$ spanned by the set $\{u_1, u_2, . . . , u_k\}$. Prove that $U\subseteq \tilde{U}$ for any subspace $\tilde{U}\subseteq V$ which contains ...
0
votes
1answer
41 views

Why isthe dimension of the vector space $M_{3\times4}$ is $3\times 4$?

I understood the concept of dimension of a vector space, which is the number of vectors in all basis of a certain vector space $V$. I understand that, for example, the $dim(\mathbb{R}^2)=2$ or ...
1
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2answers
133 views

Prove that the Eigenvalues of this Matrix are in [0,1 ]

Let $E,F \subset \mathbb{R^n}$ Note that $< . >$ defines the Inner product on $\mathbb{R^n}$ Let $(e_1,....,e_k)$ and $(f_1,.....f_l$) be Orthonormal bases of E and F respectively. Consider ...
0
votes
1answer
23 views

Prove $cof(A^t) = cof(A)^t$

I'm trying to use $A^{-1} = cof(A)^tD$, where $D = det(A)^{-1}$ to prove $cof(A^t) = cof(A)^t$. I end up with statements these two $A^{-1} = Dcof(A)^t$. And $(A^{t})^{-1} = D cof(A^t)$. But I don't ...
0
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1answer
29 views

Prove the Vector Space is Indecomposable into a Direct Sum of Invariant Subspaces

Let $T$ be a Linear Operator on a Finite Dimensional Complex Vector Space $V$ Prove that $V$ with only one Line invariant under $T$ is indecomposable into a Direct Sum of nonzero subspaces invariant ...
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2answers
36 views

question about the proof that the set C[a,b] with uniform norm is complete

I am trying to understand the proof that the set of continuous function is complete under uniform/supremum norm. First, suppose we have a Cauchy sequence of continuous functions ${f_n(t)}$ with ...
1
vote
1answer
49 views

Why in most of exercise of Linear Algerbra field involved is a subfield of complex numbers

I am studying Linear Algebra by Hoffman , they have written that reader should assume that field involved is a subfield of complex numbers , they have explained a reason beyond this by giving the ...
0
votes
1answer
63 views

Help me with this.

Let $a=2i -j+ k$, $b =i +2i- k$ and $c= i + j -2k$ be three vectors. If the vector $d$ is on the plane of $b$ and $c$ and its projection on $a$ is of magnitude $\sqrt{2/3}$, find $d$ later: So I know ...
4
votes
2answers
101 views

Good article (or book) about coordinate-dependent linear algebra, for those already familiar with coordinate-free aspects.

I have a decent understanding of coordinate-free linear algebra. For example: (not-necessarily-finite-dimensional) vector spaces, linear transforms, (possibly infinite) products of vector spaces, ...
6
votes
1answer
77 views

Invariant Subspace of Two Operators [duplicate]

Let $S$, $T$ be linear operators on a finite-dimensional vector space $V$ over $\mathbb{C}$. Suppose $$S^2 = T^2 = I.$$ Show that there exists either a $1$-dimensional or $2$-dimensional ...
0
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1answer
32 views

Is the lagrange interpolation polynomials a linear functional?

Im taking numerical analysis and abstract algebra and I think that the Lagrange interpolation polynomials is a linear functional, I did notice that such polynomials are in the dual basis, since it ...
0
votes
1answer
26 views

What is the dimension of the null space of A?

If A is a 7-by-5 matrix with Rank(A) = 2, what is the dimension of the null space of A? Justify your answer. Not really sure of the relationship between these things, can anyone help?
1
vote
1answer
52 views

Determine the basis for a vector space $V$?

Problem: I have 5 vectors $v_1, v_2,...,v_5$ each of them having $5$ components: $v_1 = \left[\begin{matrix} 5 \\ 4 \\ 3\\ 2 \\ 1 \\\end{matrix}\right] $ $v_2 = \left[\begin{matrix} -1 \\ 2 \\ 0 ...
0
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0answers
12 views

Does it make any difference how you set up a matrix when finding a basis?

If finding a basis using this method here https://www.youtube.com/watch?v=0utd-Noc_Fs Does it matter whether you set up the matrix with the vectors along the columns of the matrix as in the video or ...
0
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0answers
52 views

How To Determine A Subspace of Mnn

Using Theorem 4.2.1, which states: If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the following conditions are satisfied a. If u and v are vectors ...
0
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0answers
25 views

Find the change of basis matrix using ordered basis and standard basis?

Let B = {(3, −1), (2, 4)} be a (ordered) basis for R2, and let E = {(1, 0), (0, 1)} be the standard basis. (a) Find the change-of-basis matrix P E←B I'm not sure how to use the standard basis to ...
1
vote
1answer
22 views

Question about determining a basis for a vector space $V$.

Okay so say I want to find a basis for a vector space $U$ which is a subspace of $\mathbb{R^3}$. Where $V$ is a spanning set of $U$ and $$V=\{[-1,3,1],[0,1,3],[-1,2,-2],[1,2,14]\}$$ my question is ...
1
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0answers
19 views

Show that the set of continuous functions on $[0, 2\pi]$ follows the dot product associative property

$$f \cdot g = \int_0^{2\pi} f(t)g(t) \, dt$$ Prove that $$f \cdot (g \cdot h) = (f \cdot g) \cdot h$$ I'm most just confused as to how to deal with the $\int_0^{2\pi} g(t)h(t) \, dt$ I don't know ...
0
votes
2answers
8 views

Equation of a plane that crosses the axes at points equidistant from the origin.

Give the equation of a plane that crosses the axes at points equidistant from the origin. Explain your reasoning. I know the equation should be on a 45 degree angle looking towards the axis. I have ...
0
votes
1answer
25 views

Coplanar Vectors Proof

I came across this question in a math textbook: Prove that the vectors a=3i+j-4k, b= 5i-3j-2k, c= 4i-j-3k, are coplanar. This was my attempt at a solution: If (a x b) x c = 0, then c is orthogonal ...
0
votes
1answer
41 views

How do I check if a set is a vector space using axioms?

A. Check that $\left\{ \begin{pmatrix} x \\ y\end{pmatrix} \middle\vert\; x, y \in \Bbb R \right\} = \Bbb R^2$ (with the usual addition and scalar multiplication) satisfies all of the parts in ...
0
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1answer
23 views

Calculate intercept location and time

I want to calculate the coordinates and time where object B2 can intercept object B1. Asuming the following: All coordinates are in pixels B1: Moves from coordinates A1 to A2, it has a speed of 2 ...
0
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2answers
31 views

How to determine the dimension of a row space.

Okay so I'm doing a question where first it asks you to state a row space of a matrix and then find the dimension of this row space. I have the row space as ...
0
votes
2answers
26 views

Which of the following sets of $n \times n$ matrices with real entries is a vector space over $\mathbb R$?

Which of the following sets of $n \times n$ matrices with real entries is a vector space over $\mathbb R$ if the vector addition and scalar multiplication are as usual? A) the set of invertible ...
0
votes
1answer
15 views

Converting General equation to parametric and vector.

If the equation of my plane is (x−1,y−5,z−6)⋅N⃗ =0 What can I do to convert it to Parametric equation and vector equation?
0
votes
1answer
16 views

Finding parametric, general and vector equations of a plane given $3$ points.

Find the vector, parametric and general equations of the plane through the points $(1, 5, 6), (-3, 5, 4)$ and $(2, 1, 3)$.
0
votes
1answer
22 views

Finding the equations of two lines that meet in a point.

"Give the equations of two lines that meet at the point (2, -3, 5) and which meet at right angles, but do not use that point in either of the equations." I am having a bad time with this one. I ...
1
vote
1answer
102 views

Optimization over vector spaces. Generalized KKT.

I am looking for the extension of the theorem I found in the book by Luenberger called "Optimization by vector space methods." Here is the statement of that theorem from Luenberger: Generalized ...
0
votes
2answers
39 views

How to find the vector equation of a plane given the scalar equation? [closed]

How would I find the vector equation of the plane: $x + 2y + 7z - 3 = 0$ So far, I found the normal vector: it's $(1, 2, 7)$.
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2answers
110 views

Trivial Property of vector spaces

In several Linear Alegebra textbooks there is always a property of vector spaces listed that seems pointless to mention given it can be simplified. Can someone please tell me the significance of it ...
0
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0answers
68 views

Property of general vector space

Let $V_n$ be a finite dimensional vector space with a norm without scalar product. Let $v_i$ $(i \in \{1,\dots,n\})$ be the base of $V_n$. For each vector $w \in V_n$ there exist $x \in E_n$ ...
0
votes
1answer
17 views

plane angle calculation problem

in calculating the angle between the plane $2x + y -2z +4 = 0$ and $z$ axis I got that the angle between the normal and $z$ axis is $131.81$. however if I take $90°$ minus that I get a negative angle ...
1
vote
2answers
35 views

Vector spaces and linear dependence?

I'm struggling with this question: Find $\alpha$ such that the set of vectors is linearly dependent. $$\begin{bmatrix} 1\\ 2\\ 7 \end{bmatrix},\begin{bmatrix} 2\\ 11\\ -5 ...
1
vote
0answers
67 views

Can we say anything about the relationship between these functors?

I am working with a category $\mathcal{C}$ and two functors $F:\mathcal{C}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ and $G:\mathcal{C}^{\operatorname{op}}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ where ...