For questions about the linear span of a set of vectors, which is the smallest subspace containing the set. Most questions with this tag belong to (linear-algebra) or (functional-analysis).

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2
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1answer
22 views

Understanding the Replacement Theorem (Exchange Theorem)

I'm learning about Basis and Spans and now that's I've figured out what these are, I'm trying to understand the Replacement Theorem(also called the Exchange Theorem). The definition goes like this: ...
1
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2answers
23 views

direct sum of vectors

$$U = \{(x,y,z,t) \in \mathbb{R}^4 | x + 5y + 4z + t = 0 , y + 2z + t = 0 \} $$ $$W = \{(x,y,z,t) \in \mathbb{R}^4 | x + z + 3t = 0, 2x-3y-4z+3t = 0\} $$ $U \oplus W = \mathbb{R}^4$? This is my ...
2
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3answers
36 views

Understanding the difference between Span and Basis

I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for. I understand that the Span ...
0
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0answers
13 views

how to find out how many minimum spanning trees does a graph have [on hold]

so i was wondering what method can someone use to find the amount of MSTs in a graph Thanks in advance
1
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1answer
15 views

Finding a base from matrice subspace

$U,W$ are sub-spaces of $M^\mathbb{R}_{2x2}$ $$U=Sp\left\{\begin{pmatrix} 1 & 2\\ 4 & 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & -1\\ 3 & 2 \\ \end{pmatrix}, \begin{pmatrix} 1 & ...
2
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1answer
51 views

Basis and dimension of the span of the vectors (0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)

Find a basis for the given subspace by deleting linearly dependent vectors. $S = \text{span}\{(0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)\}$ I do not understand how to "delete linearly independent ...
1
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2answers
18 views

linear independency in equation of linear span

we got the following vectors: $$v_1, v_2, w_1, w_3 \in V$$ $V$ is a vector space so that $\DeclareMathOperator{Sp}{Sp}\Sp\{v_1,v_2\} = \Sp\{w_1,w_2\}$ it's also defined that $\{v_1,w_2\}$ is linear ...
0
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1answer
32 views

Finding Projection for Non-linear constraint [closed]

I think, I was not able to ask the question clearly. Sorry for that. I will try to ask it in a different way. Suppose I want to find the orthogonal projection of (x1,x2,y1,y2) such that x1=x2, y1=y2. ...
0
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1answer
16 views

solving equation with linear span (using row reduction)

We've got the following span: $$U = Sp\{(2, 5, -4, -10), (1, 1, 1, 1), (1, 0,3,5) , (0,2,-4,-8)\}$$ We need to find the values of the number $a$ where the vector $$v = (a, a-6, 4a-3, 6a-1)$$ belongs ...
2
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1answer
46 views

A set is linearly dependent if and only if there is a proper subset with the same span?

Let $S$ be a subset of a vector space. I conjecture that $S$ is linearly dependent if and only if there exists a proper subset $S' \subset S$ such that ...
0
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1answer
33 views

understand an answer to linear span of polynomal subspace exercise

i am looking at an answer to an exercise who asks to find a linear span for, and I don't really understand the solution $$p(x) = ax^3 + bx ^2 + cx + d$$ and this is the solution i see $$ p(x) \in M ...
1
vote
1answer
31 views

find equality between linear spans

$$U = Sp\{(2,5,-4,-10), (1,1,1,1),(1,0,3,5), (0,2,-4,-8)\}$$ $$ W = Sp\{(1,-2,7,13), (3,1,7,11), (2,1,4,6) \}$$ two questions: prove that $U = W$ find the values of the $a \in \mathbb{R}$ where the ...
0
votes
2answers
19 views

linear span of subspace

we have the following subspace over $\mathbb{R}$ $$M = \{ A \in M^{{n\times n}} | A = -\overline{A} \}$$ I found that it is a subspae and now I need to find the linear span of it. how can I calculate ...
0
votes
1answer
29 views

Is a linear span of finite set from a finite dimensional space topologically closed?

Let $S=\{x_1,\ldots,x_m\} \subset \mathbb{C}^n $ is it true that: $$ Span (S) = \overline{Span (S)} $$ Must we assume both of the following assumptions? or one of them will be enough? The spanning ...
1
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2answers
51 views

Basis of a subspace (What does it mean for vectors to span a subspace?)

How is the basis of this subspace the answer below? I know for a basis, there are two conditions: The set is linearly independent. The set spans H. I thought in order for the vectors to span H, ...
2
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1answer
31 views

If $v_1,…,v_m$ are linearly independent, then the span $v_1+w,…,v_m+w$ has dimension $\ge m-1$

Suppose $v_1,...,v_m$ is linearly independent in $V$ and $w\in V$. Prove that $$ \dim (\operatorname{span}(v_1+w,...,v_m+w)) \ge m-1$$ It's an exercise in the book Linear Algebra Done Right. ...
1
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3answers
26 views

Determine whether the vectors span $\mathbb{R}^3$

They want us to determine whether it span $\mathbb{R}^3$ and they gave us these vectors $V_1=(1,2,6), V_2=(3,4,1), V_3=(4,3,1), V_4=(3,3,1)$ and the answer is that the vectors span $\mathbb{R}^3$ . ...
0
votes
1answer
27 views

$Span(A)\cap Span(B\setminus(A\cap B))=\{\vec 0\}\Longrightarrow Span(A)\cap Span(B)=Span(A\cap B)$?

$A$ and $B$ are two linearly independent sets. $A\cap B = \varnothing$ and $A\nsubseteq B$,$B\nsubseteq A$. Is the following statement true?: $$Span(A)\cap Span(B\setminus(A\cap B))=\{\vec ...
2
votes
2answers
47 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
3
votes
2answers
103 views

Find basis and dimension of a subspace

Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by: \begin{bmatrix} 1 & -5 \\ -4 & 2 \\ \end{bmatrix} \begin{bmatrix} 1 & 1 \\ -1 & 5 \\ ...
0
votes
1answer
20 views

By defining linear independence and span, explain what it means to be a basis of V

By defining linear independence and the span $\langle S\rangle$, explain what it means to say that $S$ is a basis of $V$. (3 marks) I'm not entirely sure if I've got this correct so I'm going to give ...
0
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1answer
45 views

Linear Independence and “Not in the Span”

I'm studying elementary linear algebra right now, and the current section is on linear independence. As I create matrices from the vectors and row reduce them in a calculator, I get various results. ...
0
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1answer
22 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 ...
3
votes
3answers
212 views

Visualization of span of 3 vectors?

If I visualize 3 vectors in 3d space, is the span of the 3 vectors a 3d region within the 3 vectors? I need help visualizing so I can understand the concept better.
1
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0answers
51 views

Dimensions of spaces over different fields

We view $\Bbb C_2 = \{{w \choose z}:w,z\in \Bbb C\}$ as a vector space over $\Bbb C$, $\Bbb R$ and $\Bbb Q$. Let $x_1={i \choose 0}$, $x_2={\sqrt2 \choose \sqrt5}$, $x_3={0 \choose 1}$, $x_4={i\sqrt3 ...
0
votes
1answer
24 views

showing equality of dimensions

Let $\alpha \in \mathbb{C}$ be a complex number. Let $V = \mathbb{Q}(\alpha)$ be the rational vector space spanned by powers of $\alpha$. That is $V = <1,\alpha,\alpha^2,\ldots>$. If $P(t)$ ...
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0answers
31 views

hamming weight of error correcting codes and BCH codes

In general the hamming weight of codewords of error correcting codes is well understood. If I were to write down the $k \times n$ generator matrix, with the span of the rows corresponding to the ...
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2answers
27 views

Showing that a dual basis is a generating set

Suppose we have a dual basis $F$* = ($f_1$*,......$f_n$*) of $V$*. and suppose the standard basis $F$ = ($f_1$,......$f_n$) of $V$. I want to show that the $F$* is a basis so i have to show that it is ...
0
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2answers
29 views

Linear Algebra: Does it span? please correct me

Can anyone please correct me? my problem is in the proof part below Q: Does {(1, -1),(2, 1)} spans R2? A: c1(1, -1) + c2(2, 1) = (x, y) c1 + (2)c2 = x -c1 + c2 = y ______________ c1 = x - ...
0
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2answers
60 views

How to determine if a set of vectors is a basis for a subspace?

So I have a homework question which I am not sure if I am answering correctly. The questions is as follows. Determine whether the set is a basis for $\mathcal{R}^3$. If the set isn't a basis, ...
0
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2answers
53 views

How do I check if a vector space spans a particular vector?

I have been given $5$ vectors $(u_1, u_2, u_3, u_4 \text{ and } u_5)$: $u_1= \langle1,-1,2,1\rangle, u_2 = \langle1,2,1,-1\rangle, u_3 = \langle-1,-8,1,5\rangle, u_4 = \langle1,1,1,1\rangle, u_5 = ...
0
votes
0answers
19 views

probability of vector in column span

Consider we have a fixed matrix M of size a$\times$ 2b (Let us look at M=[$M_1$ $M_2$] where matrices $M_1$,$M_2$ are of size a$\times $b) and a vector $v$ of dimension a. Is there any way that I can ...
-1
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3answers
39 views

If $\dim(V) = n$, is every spaning set $\{v_1,v_2,\ldots,v_n\}$ a basis for $V$?

Okay, so I need help clearing things up. Let $V$ be a vector space and $dim(V)=n$. Does it mean that every Spanning set $\{ v_1,v_2,v_3,\ldots,v_n \} $ is necessarily a basis for V? ...
3
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2answers
78 views

For what values of k is the vector u in the span of the columns of A?

Let $$u = \begin{pmatrix} -1 \\1 \\-1 \end{pmatrix}$$ and let$$A = \begin{pmatrix} 1&0&1 \\1&1&0 \\0&1&k \end{pmatrix}$$ For what values of k is the vector u in the span of ...
2
votes
1answer
60 views

If we have a square matrix thats invertible, do its row and column space coincide?

If we have a square matrix thats invertible, do its row and column space coincide? Regarding an nxn invertible matrix: -The row space of the matrix is R^n -The column space of the matrix is R^n ...
0
votes
1answer
21 views

Span and Linearly independence of a set

Suppose that $(V, +, \cdot)$ is a vector space over a field $F$ and $S = \{v_1, v_2, \ldots, v_k\}$ is a subset of $V$. Describe the span of $S$. Explain how to determine whether $S$ is linearly ...
0
votes
1answer
76 views

How to find a subset that contains all linearly independent polynomials?

I found that a set S is linearly independent. How can I find a subset A of S that contains all linearly independent polynomials? My set S consists of the following polynomial vectors in P3: pv1 = ...
0
votes
1answer
41 views

Let W be an infinite dimensional vector space.Under what conditions are there only a finite number of distinct subsets S of W such that S generates W?

let W be a subspace of a vector space V. Under what conditions are there only a finite number of distinct subsets S of W such that S generates W? If W is finite then obviously there only a finite ...
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2answers
59 views

Find a vector that spans the given set

Question in book: Let H be the set of all vectors of the form [-2t, 5t, 3t]. Find a vector v in R3 such that H=Span{v}. Why does this show that H is a subspace of R3? Answer from solution ...
5
votes
3answers
263 views

what does the set containing only the zero vector actually span?

I apologize if this sounds stupid but I am struggling to grasp the following concept. I understand that the span of the empty set is the zero vector. However, what does the set only containing the ...
0
votes
1answer
42 views

Does $B = \{x-2, x(x-2), x^2(x-2)\}$ span $\{p(x)\in P_3(\mathbb{R})|p(2) = 0\}$?

Let $P_3(\mathbb{R}) = \operatorname{Span} \{1, x, x^2, x^3\}$. $W$ is a subspace of $P_3(\mathbb{R})$, $W = \{p(x)\in P_3(\mathbb{R})|p(2) = 0\}$. Find a basis and the dimension of $W$. I chose ...
1
vote
0answers
56 views

Linear algebra: - Linear independence and span

I need to know find all values of the parameter $t$, such that vectors $v, u, w$ span $\Bbb{R}^3$ $u= (1,1,1)$ $v=(1,5t,1)$ $w= (1,1, 25t^2)$ b) for which parameter $t$ is the set linearly ...
1
vote
2answers
33 views

Prove that $\{v_1,v_2…v_n\}$ is linearly dependent if $a\notin span\{v_1,v_2,v_3…v_n\}$ and $v_n\in span\{v_1,v_2…v_{n-1},a\}$.

I ran into the next problem and got really confused: Let $\{ v_1, v_2,v_3... v_n \}$ be a set of vectors in the vector space $V$, and let $a\in V$ in such a way that: $a\notin ...
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0answers
87 views

If a set of $n$ vectors spans an $n$-dimensional space, it is linearly independent.

Prove that in an $n$ dimensional vector space $V$, a subset $S$ of $ V$ which has $ n$ vectors and $L(S)=V$, i.e.linear span of $S$ is $V$, is linearly independent. I can solve the problem by ...
1
vote
2answers
44 views

Best way to find rref of a matrix?

I'm sitting here doing rref problems and many of them seem so tedious. Any tricks out there to achieve rref with less effort or am I stuck with rewriting the matrix for every 2/3 operations? I know ...
0
votes
2answers
42 views

How do I verify that a set of vectors is a basis for the given plane

I have a set of 2 vectors: $\{ (1,2,0), (0,2,-1) \}$. I have to show that this set is a basis for the plane with equation: $2x_1 - x_2 -2x_3 = 0$. I know that the normal vector of the plane is ...
0
votes
3answers
36 views

for which $a$ values will $U$ and $V$ span the same space

having trouble with this question... if anyone could give me a hint on where to begin that would be lovely. Let the set of vectors $U= \{u_1,u_2\}$ $\space$ when $u_1 = (1,a,2)\space $and$\space ...
0
votes
0answers
26 views

Find images of 2 x 4 and 4 x 2 matrices

I'm working on some basic linear algebra stuff and I'm hitting a roadblock. I have a prompt that says "for each matrix A, find vectors that span the image of A. Give as few vectors as possible." I ...
2
votes
3answers
41 views

Show that a vector $\langle h,k\rangle$ is in span${u, v}$ for all $h$ and $k$

I have a question with two vectors and it asks to prove that a third vector is in the span of the two vectors. Let $u = \left[ {\begin{array}{c} 2 \\ -1 \\ \end{array} } \right]$ and $v = ...
1
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3answers
288 views

Span of the trivial vector space

I was wondering is the span of the trivial vector space the {0} vector in $R^2$ for example just 1 vector (0,0) or can it be any vectors in R^2 since we can pick $t \in R$ to be zero so in this case ...