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).
Questions tagged [span]
177 questions
25
votes
3 answers
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 of a Vector Space $V$ is the linear combination of…
nTuply
- 701
14
votes
5 answers
Span of an empty set is the zero vector
I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone".
Is Nering defining the span of the empty set to be the set…
Matt Brenneman
- 1,522
11
votes
5 answers
Why do we need "span" in linear algebra?
In my linear algebra course in university we started learning about span and I was curious what is it good for? and if someone know, how does it relate to 3D graphics?
Thank you.
LiziPizi
- 2,905
10
votes
3 answers
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 zero vector span? The zero vector as well? Also, are…
ponderingdev
- 576
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- 19
5
votes
5 answers
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 $u_1, u_2, . . . , u_n \in V$ are linearly…
Jess
- 125
4
votes
3 answers
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 but that will be time consuming and will most…
Amuna
- 377
4
votes
3 answers
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 column in $B$ to obtain a new basis for $V$?
My…
Bill DeRose
- 144
4
votes
3 answers
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 is no (I thought you needed n vectors to span…
user124145
4
votes
2 answers
Understanding Replacement Theorem and Linear Dependence Lemma
I'm reading Axler's Linear Algebra Done Right and am hung up on one part of Axler's Replacement Theorem proof.
His proof states the following:
"Suppose $u_1,...,u_m$ is linearly independent in V. Suppose also that $w_1,...,w_n$ spans V.
...the…
pomelozest
- 140
- 8
4
votes
4 answers
Overview of Linear Algebra
Very new student tackling this course, and I've never been this terrified from Math before. I cannot grasp the meaning of things in Linear Algebra, most of what's stated is either obscure, meaningless, and abstract when I am first tackling them.…
AAS.N
- 291
4
votes
1 answer
What does "span" looks like in infinite dimensional spaces?
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 vector spaces?
DO we represent this using integral or…
Your neighbor Todorovich
- 8,472
4
votes
2 answers
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.
I'm wondering if I can write $U_1…
When
- 571
4
votes
0 answers
$\rm span(S_1) + \rm span(S_2) = \rm span(S_1 \cup S_2)$ for infinite sets
I have these two definitions of span:
Span:
Suppose a vector space $(V,+,\cdot)$, and
$$S = \{u_1,\cdots,u_n\}$$
(and $S$ is a subset of $V$, not a subspace)
$$[S]=:\cap_{w\subset V, w\supseteq S} W$$
In other words, $[S]$ is, by definition, the…
Poperton
- 6,594
3
votes
2 answers
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 = \begin{bmatrix}-1\\k\\7\end{bmatrix}, \quad
a_2 =…
Michael0x2a
- 563
3
votes
1 answer
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 $\text{Span}\left\{\mathbf{a},\mathbf{c}\right\}$.
Why is it false? Have I forgotten to consider…
Ming-Tang
- 357