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7 views

Number of pivot columns in a 4x6 matrix for spanning set to occur

How many pivot columns must a 4x6 matrix have if its columns span $\mathbf{R}^4$? Explain. So, in my head, this is pretty clear: You need four dimensions => So you need a minimum of four vectors that ...
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1answer
24 views

Do the given vectors span $\mathbb{R}^3$?

Do the following vectors span $\mathbb{R}^3$: $$v_1 = (2, -1,3)$$ $$v_2 = (4, 1, 2)$$ $$v_3 = (8, -1, 8)$$ I use Gaussian Elimination to bring the matrix to an echelon form, with a pivot of "1" in ...
0
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0answers
11 views

Conditional independence of span

The span of a matrix is just the linear combination of its vectors. Let the bolded ${\bf Z}$ denote a random variable matrix and $X,Y$ be random variables. Would the following be true? $Y \perp X | ...
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3answers
51 views

Find a vector in the matching dimension that is not in the span

I have the following vector $(1,2,-2),(2,-1,1)$. How do I find a vector that is not in the span of those two vectors. I can pick an arbitrary third vector and make the other two vectors equal to it ...
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2answers
47 views

Proof that $[v, Tv, T²v, … , T^n v]$ is a basis for $V$ ($dim(V)=n$)?

Let $T:V\to V$ be a linear map from a finite dimensional vector space over a field $F$ to itself. Assume $[v,Tv,T²v,...]$ spans $V$ for some $v \in V$. Don't know at all how to prove that $[v, Tv, ...
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2answers
37 views

Find the value(s) of $k$ such that the given vectors do not span $\mathbb{R}^3$

I'm currently attempting to solve the following problem: Find the value(s) of $k$ such that the vectors $\{\vec{a}_1, \vec{a}_2, \vec{a}_3\}$ do not span $\mathbb{R}^3$, where: $$ a_1 = ...
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4answers
62 views

Can linearly dependent set span?

Can a linearly dependent set A = {(1,0,0), (0,1,0), (0,0,1), (1,2,3), (3,4,5)} span? Since columns 4 and 5 are linear combinations of 1,2 and 3, would spanA equals the span of columns 1,2, and 3? ...
3
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1answer
28 views

Why is this statement about $\text{Span}$ false?

Here is a true-false question known to be false: If $\mathbf{a}$ is in $\text{Span} \left \{ \mathbf{b}, \mathbf{c} \right \}$, then $\mathbf{b}$ is in ...
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0answers
10 views

Proof that the span of a list is equal to the span of any reordering of the list

Claim: If $(w_1,w_2,...w_m)$ is an arbitrary reordering of $(v_1,v_2,...v_m)$, then $span(w_1,w_2,...w_m) = span(v_1,v_2,...v_m)$. Proof By definition, ...
1
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2answers
94 views

Find values of h such that the vectors (2, 4) and (h, 6) span $\mathbb{R}^2$

My homework is asking me to answer problems such as the one that follows: Find all values of $h$ such that the vectors $\{a_1, a_2\}$ span $\mathbb{R}^2$, where $a_1 = (2, 4)$ and $a_2 = (h, 6)$. I ...
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1answer
38 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
3
votes
2answers
23 views

A real matrix whith rows generating $U$ and columns generating $V$

Let $n \in \mathbb{N}$, and $U,V$ two linear subspaces of $\mathbb{R}^n$ of the same dimension. Could one always make a matrix $A \in \mathbb{M}^{n \times n}(\mathbb{R})$ such that $spanA = U$ and ...
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0answers
12 views

For what values of λ is this family free (independent), spanning and a basis of R[t]≤3

The family of polynomials $F$ = {${(λ^2 − 1)t^3 + t^2, λt^3 + t − λ, (1 − λ)t^3 + t + 1, λ}$} in $R[t]_{≤3}$ I set their sum to 0 to find the values for it to be independent. $a((λ^2 − 1)t^3 + t^2) ...
1
vote
1answer
38 views

Determine if the following is a subspace and find its smallest possible subspace of $\mathbb{R}^3$

$U_k = \{(x_i)_{1≤i≤n} \in \mathbb{R}^n\ |\ x_k = 0\}$. Is this a vector subspace of $\mathbb{R}^n$? For $n = 3$, what is the smallest vector subspace of $\mathbb{R}^n$ that contains $U_1, U_2, U_3$. ...
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3answers
98 views

Find a value r so that the vector v is in the span of a set of vectors

Find the value r so that, $$v = \begin{pmatrix} 3 \\r \\-10\\14 \end{pmatrix}$$ is in the set, $$ S= \text{span}\left(\begin{pmatrix} 3\\3\\1\\5 \end{pmatrix}, \begin{pmatrix} 0\\3\\4\\-3 ...
0
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1answer
34 views

Why do these vectors not span the given space?

I need some help understanding this solution to a problem. I am working on the problem above. I know that in order for a set of vectors to be a basis it must be linearly independent and span the ...
0
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0answers
90 views

What is the difference between the span of a set to its subspace?

I am confused with some of the definitions of linear algebra. I know that the span of set S is basically the set of all the linear combinations of the vectors in S. The subspace of the set S is the ...
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2answers
43 views

i am having trouble with one of the homework question regarding to linear algebra(vector and span)

$V$ is a vector space of some dimension, with $\vec u,\vec v,\vec w$ independent set of vectors in $V$. define the subspace of $V$ given by $W = \operatorname{span}(\vec u-\vec v+\vec w, 2\vec u+\vec ...
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3answers
89 views

does the set of vectors $S =\{u,v,w\} = \{(1,2,3),(2,2,1),(0,4,-5)\}$ span $\mathbb R^3$? If not, What does it span? Describe it geometrically.

does the set of vectors $S=\{u,v,w\}=\{(1,2,3),(2,2,1),(0,4,-5)\}$ span $\mathbb R^3$? If not, What does it span? Describe it geometrically. I already know I'm supposed to turn this into a $3\times3$ ...
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2answers
43 views

Calculate the dimension of $U = \{(x_1,x_2,x_3,x_4,x_5) : x_1+x_3+x_5=x_2+x_4=0\}$

In the vector space $V \subset \Bbb R^5$, considering the vectors $v_1,v_2,v_3$ $v_1 = (0,1,1,0,0)$ $v_2 = (1,1,0,0,1)$ $v_3 = (1,0,1,0,1)$ We have $V = \mathrm{span}(v_1,v_2,v_3)$ ...
0
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1answer
34 views

How to complete a span

Hey I have a simple question. Say I have vector space V (dimV > 1). B = Span{v1,v2} and C = span{u1}. I have to find a vector u2 such that span{u1,u2} = span{v1,v2} = B. How do I do it ? thanks
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3answers
57 views

Why does linearly independent spanning set imply minimal spanning set for a vector space?

Suppose β is a linearly independent spanning set of some vector space V. Why must it be the minimal spanning set? In other words, why can there not be two linearly independent spanning sets of a ...
0
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3answers
31 views

find a base to U Linear Algebra

dear users please help me... im answering a long question now ive been guided to find a base to U at the end of the process i got this $u= Sp\{x^4-3x^3+2x^2, 3x^4-7x^3+4x ,1\}$ and ive been guided to ...
0
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1answer
76 views

Show that S (a subset of V) is contained in span(S)

Let $\text{span}(S) = \lbrace v \in V \mid v\ \text{is a linear combination of vectors in}\ S\rbrace$. I need to show that $S$ is contained within $\text{span}(S)$. I know if $S$ is nonempty, $0$ is ...
0
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1answer
71 views

Basis of M2,2 is not spanning set of trace zero matrices?

given set of matrices: S={[1 0; 0 0];[0 1; 0 0];[0 0; 1 0];[0 0; 0 1]} I have to explain why S is not a spanning set of matrices with trace zero, matrices of: V be the subspace of M2,2: V = {[a b; ...
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0answers
21 views

functional analysis findining dist(x,Z) in L2(-pi,pi)

The question in my hw was Let Z=span (1,sint,cost), x(t)=t. Find dist(x,Z) in $L_2(-\pi,\pi)$ From a lemma that we learned it says if Z is closed and $x(t)=t \notin Z$ then ...
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4answers
628 views

How to prove a set of vectors does not span a space.

Ok, so I'm a bit curios as to how you can prove a set does not span a vector space. For example, let ${S}$ be the vector set \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0\\ ...
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3answers
42 views

You can only take the span of linearly independent vectors?

Ok, this might be a bit trivial but I'm having trouble wrapping my head around my text book. So, to my understanding for ${Span(v_{1},v_{2},..,v_{n})}$ then ${v_{1},v_{2},..,v_{n}}$ must be linearly ...
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3answers
51 views

Is it possible for a set of non spanning vectors to be independent?

I was reading about linear spans on Wikipedia and they gave examples of spanning sets of vectors that were both independent and dependent. They also gave examples of non spanning sets of vectors that ...
1
vote
1answer
67 views

Finding a basis for span of vectors

$U = \text{span}\{(1,0,0),(0,2,-1)\}$, $W = \text{span}\{(0,1,-1)\}$. How can I find bases for $U$ and $W$? (I think they're linearly independent, right?) Can I just take $B_1 = \{(1,0,0),(0,2,-1)\}$ ...
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2answers
117 views

Find the closest point p in S to the point w, given

NOTE: Nobody showed me how to do this before. I AM DESPERATE for a step by step solution. Please help!! Let $S$ be the subspace of $\mathbb R^3$ spanned by vectors $u$ and $v$. Find the closest point ...
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3answers
52 views

Determine whether each given sequence…

spans $M_{2\times 2}$ is a basis for $M_{2\times 2}$ $$ \mbox{(a)} \qquad \left( \begin{pmatrix} -4 & 1 \\ 0 & 5 \end{pmatrix}, \begin{pmatrix} -3 & 0 \\ ...
0
votes
1answer
51 views

Is this vector in the span?

a. Is $(13, -12, 14, 4)^T$ in $\mathrm{Span}\{(-4, 3, 2, -2)^T, (-2, 2, -3, 4)^T, (5, -4, 0, 4)^T\}$? b. Is $1 + x^2$ in $\mathrm{Span}\{-4 + 3x + 2x^2 - 2x^3, 5 - 4x + 4x^3\}$? c. Is $(15, -14, 18, ...
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2answers
88 views

Linear Algebra Explanations on true and false.

1.Could someone prove that if a set of vectors in a $p$-dimensional vector space $Q$ is a spanning set for $Q$, it is a basis. 2.If $T$ is a linear transformation from $\mathbb R^3$ onto $P_2$, then ...
1
vote
2answers
113 views

Show that a set of vectors spans $\Bbb R^3$?

Let $ S = \{ (1,1,0), (0,1,1), (1,0,1) \} \subset \Bbb R^3 .$ a) Show that S spans $\Bbb R^3$ b) Show that S is a basis for $\Bbb R^3 $ I cannot use the rank-dimension method for (a). Is it ...
0
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2answers
114 views

Linear Algebra Solution set equal to Span

I am confused on how to approach a certain question. It asks to find a collection of vectors $x_1,\ldots,x_p$ so that the solution set of equation $ax=0$ is equal to $\operatorname{span} \{x_1,\ldots ...
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1answer
62 views

Linear Algebra Span question

Let $a, b, c$ be vectors in $\mathbb{R}^3$. From what I understand, if $c\in \mathrm{Span}\{a,b\}$, then $b\in \mathrm{Span}\{a,c\}$. Since they all fall on the same plane, I can't seem to find a ...
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3answers
168 views

Is it possible to swap vectors into a basis to get a new basis?

Let $V$ be a vector space in $\mathbb{R}^3$. Assume we have a basis, $B = (b_1, b_2, b_3)$, that spans $V$. Now choose some $v \in V$ such that $v \ne 0$. Is is always possible to swap $v$ with a ...
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3answers
96 views

Why can't two vectors span $\Bbb R^3$?

I came across a question in my linear algebra textbook and it said: "Given $x_1 = (1, 1, 1)^T$ and $x_2 = (3, -1, 4)^T$: Do $x_1$ and $x_2$ span $\Bbb R^3$? Explain." I'm pretty sure that the answer ...
0
votes
2answers
33 views

Which set Spans the Same set

Which set spans the same set as $\left\lbrace(1,2,−1),(0,1,1),(2,5,−1)\right\rbrace$ ? The answers choices each give either a set of 3 vectors or 2 vectors. How will I go about to solve this ...
0
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1answer
60 views

Proof that two spans are equal

I have an orthogonal subset of nonzero vectors $u_1,\dots,u_n$. I take $v \in V$ (a vector space) so that $v$ is the in the span of the previous set. Now I let $u$ be $v-m_1u_1-\dots-m_nu_n$ with ...
0
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1answer
52 views

Find two linear homogenous equations that are equivalent to the span of a line in 3 dimensions.

I'm working on an exercise in linear algebra and I'm stumped by part of it. Say I have a line in 3 dimensions that can be represented as $\operatorname{Span}\lbrace[1,-2,-2]\rbrace$. The book says ...
0
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1answer
57 views

Distance from Vector to the Linear Span

Let $V$ be the space of real polynomials of degree $\leq n$. a) Check the setting $(f(x),\,g(x))=\int_{0}^{1}f(x)g(x)\,dx$ turns $V$ to a Euclidean space. b) If $n=1$, find the distance from ...