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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Calculating the Span of a Matrix in MATLAB

If I have a matrix (or a set of vectors) say A=[1 2 4] [2 9 8] [7 9 3] how can I calculate its span in MATLAB? There is no direct command for it? Do I have to form a set of linear ...
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1answer
25 views

Span of a set of k points in $\mathbb{R}$

I'm trying to understand the following proof: Given k points $p_1,...,p_k$ in $\mathbb{R}^n$. Then (for all $i,j =1,...,k$): ...
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1answer
98 views

What does “span” looks like in infinite dimensional spaces? [closed]

I noticed that my prof loves to write $S = span\{v\}$ Instead of $\sum \alpha v$ or $a_1v_1 + a_2v_2+...$. Is he using "span" in a general way? What would span look like in infinite dimensional ...
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53 views

Decide whether the vector (2,2,4,1) belongs to the span of vectors:

Decide whether the vector $(2,2,4,1)$ belongs to the span of vectors: $$(1,1,1,-1)$$ $$(-2,1,3,1)$$ $$(3,2,1,1)$$ so far I have this by using guassian elimination: Let $(2,2,4,1)$ be known as w ...
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1answer
42 views

Condition to Guarantee $n$ Distinct Eigenvalues

I looked around, and as far as I can tell, I haven't found this question anywhere else on SE, so if I somehow missed it, please pardon me. I think this is probably a standard result, but I am having ...
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33 views

Describe the span of the given vectors geometrically and algebraically

Describe the span of the given vectors geometrically and algebraically: $\pmatrix{1\\0\\-1}$, $\pmatrix{-1\\1\\0}$, $\pmatrix{0\\-1\\1}$. I have figured out that these vectors are linearly dependent ...
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2answers
49 views

Let V be a vector space of dimension n. Prove that no set of n - 1 vectors can span V.

I'm not sure I understand the question. As far as I understand it when it says vector space of dimension n, it signifies that there will be n amount of vectors; right? So basically it wants you to ...
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35 views

Suppose $U=Span\{u_{1}, u_{2} \}$ for $u_{1}, u_{2} \in U$ and $V=Span\{ v1, v2\}$ for $v_{1},v_{2} \in V$. Prove that $U+V=Span\{u1,u2,v1,v2\}$.

This is what I have so far, I don't know if this is where I stop or if there is more to prove? $$U+V = (c_{1}u_{1} + c_{2}u_{2}) + (c_{1}v_{1} + c_{2}v_{2}) = c_{1} (u_{1}+v_{1}) + ...
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27 views

Proof to show that sums of vectors spanning a vector space also span a vector space

Let vectors $v_1, v_2, and v_3$ span a vector space $V$. Show that the vectors $v_1, v_1 + v_2$ and $v_1+ v_2 + v_3$ also span $V$. How would I go about proving this? I understand that I have to show ...
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31 views

An exercise question on Hoffman's Linear Algebra

Is the vector (3,-1,0,-1) in the subspace of $R^5$ spanned by the vectors (2,-1,3,2), (-1,1,1,-3), (1,1,9,-5)? I think these vectors all live in $R^4$ instead of $R^5$ so they the answer is no, but ...
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1answer
24 views

Linear Algebra- Sums of Vector Spaces

I dont know how to prove this although intuitively I know that it is true: Let $ V $ be a finite dimensional vector space and $S$ and $T$ be subsets of $ V $. Show that $$ Sp(S\cup T) = Sp(S)+Sp(T) ...
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1answer
32 views

Span of orthogonal complement of intersection of hyperplanes

Let $V$ be finite dimensional Euclidean vector space. Let $a_1, \ldots a_m$ be nonzero vectors in $V$. Denote $H_i:=H_{a_i}$ be a hyperplane orthogonal to $a_i$. Set $X=H_1 \cap H_2 \cap \ldots \cap ...
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3answers
64 views

Given a subset $S$, is there any universal construction (property) of $L(S)$, the linear span of $S$?

Given a subset $S$ of a vector space $V$, is there any universal construction (property) of $L(S)$, the linear span of $S$ (like the way we can construct the fraction field of an integral domain)?
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1answer
31 views

Determining span, is there an easier way to remember it?

As I understand it, to set up a problem to determine if the vector spans $ \mathbb{R}^n$ or if the given vector is in the Span of $$(v_1,v_2,...,v_n)$$ you take the vector and set up an augmented ...
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5answers
240 views

If $n=\dim(V)$ and $n$ vectors are linearly independent, then they form a basis

If $V$ is a vector space and $v_1, v_2, . . . , v_n \in V$ span $V$, and $u_1, u_2, . . . , u_m ∈ V$ are linearly independent, then $m\le n$. Use this to prove that if $V$ has dimension $n$ and $u_1, ...
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1answer
30 views

Span - linear algebra

I'm having some trouble in solving some exercises related to vector spaces, and I can't even start the solution. I need to check if the sets given span the same subset of the vector space $V$: (i) ...
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1answer
33 views

What's the difference between linear span and linear transformation?

I tried to google both definitions. For linear span, click http://en.wikipedia.org/wiki/Linear_span For linear transformation(wiki takes it as linear map), click ...
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3answers
50 views

Linear Dependent Span

$\{x \cos x, x, \cos x \}$ is a subspace of $V$. I need to find if it's a linear dependent or linear independent. So I thought that its dependent since $x \cos x$ is multiplication of $x$ and $\cos ...
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1answer
37 views

How to find the span for a linear transformation?

I'm learning Linear Transformations and I understand what is a linear transformation. Now I'm trying to look at an example question and I'm not really sure how the span is found. The question goes as ...
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1answer
39 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: ...
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2answers
30 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 ...
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44 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 ...
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1answer
17 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 & ...
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1answer
78 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 ...
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2answers
22 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 ...
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1answer
22 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 ...
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1answer
49 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 ...
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1answer
34 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 ...
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1answer
36 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 ...
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22 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 ...
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1answer
37 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 ...
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2answers
66 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, ...
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1answer
34 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. ...
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3answers
33 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 $\mathbb{R}^3$ . ...
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1answer
29 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 ...
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2answers
48 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
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229 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 \\ ...
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1answer
27 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 ...
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1answer
49 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. ...
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1answer
23 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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3answers
256 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.
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54 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 ...
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1answer
26 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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39 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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31 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 ...
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34 views

Does {(1, -1),(2, 1)} spans R2? 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 - ...
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2answers
70 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, ...
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2answers
57 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 = ...
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0answers
20 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 ...
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3answers
42 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? ...