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).

learn more… | top users | synonyms

3
votes
2answers
77 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
18 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
votes
1answer
41 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
votes
1answer
20 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
189 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
vote
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
23 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 $P(t)$ ...
0
votes
0answers
22 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 ...
1
vote
2answers
25 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
votes
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
votes
2answers
50 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
votes
2answers
52 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
18 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
votes
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
votes
2answers
71 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
58 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
18 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
68 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
38 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 ...
1
vote
2answers
51 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
182 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
36 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
54 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
31 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 ...
1
vote
0answers
83 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
37 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
37 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
21 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
38 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
vote
3answers
265 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 ...
0
votes
1answer
49 views

matrix vs vector span {} linear algebra

I am in a University Linear Algebra course and am confused by the term span and its relation to both matrices and vectors. Can someone help clarify what they mean? =Span= Can it only be made of ...
2
votes
2answers
32 views

Linear transformation with special properties

how should I do that please (I had this in my test yesterday)? Linear transformation $f:\mathbf{R}^{10} \to \mathbf{R}^7$ has an attribute that every vector $\mathbf{v}$ for which is true that ...
1
vote
0answers
37 views

How to determine if a set represents a line, plane or hyperplane?

How do you approach a question that gives you a set and asked to determine if it represents a line, plane or hyperplane? The Question: https://www.dropbox.com/s/0gscqur18kqg3ma/SpanningQuestion.PNG ...
2
votes
2answers
68 views

Prove linear independence of a set $\{\mathbf{x}-\mathbf{x_1},\ldots,\mathbf{x}-\mathbf{x_n}\}$

Let $V$ be a vector space and suppose that $\{\mathbf{x_1},\ldots,\mathbf{x_n\}}$ is a linearly independent subset of $V$. If $\mathbf{x} = \sum_{i=1}^n c_i\mathbf{x_i}$ where each $c_i \in ...
0
votes
1answer
19 views

Let $X\subseteq R^3$ That defined by : $X=\{(x_1,x_2,x_3)|x_1^2+x_2^2-x_3^2=1\}$ Find $Sp(X)$, dim(X), and find basis for $X$

I have this problem. Let $X\subseteq R^3$ That defined by : $$X=\{(x_1,x_2,x_3)|x_1^2+x_2^2-x_3^2=1\}$$ Find $Sp(X)$, dim(X), and find basis for $X$ My solution : Let $x_1,x_2 \in R$ ...
0
votes
2answers
18 views

Prove there is a linear transformation

I would like some advice with one problem. Let $L$ be a linear space with $\dim(L)=n$. Let $K$ be some linear subspace of $L$. I need to prove there exists a linear transformation $l: L \rightarrow ...
0
votes
2answers
45 views

How to prove that a set spans a plane

How do we prove that a set of vectors span a plane in $\mathbb{R}^3$? (This is not the question I am asking for help with! This is an example of the method my teacher has given us to show that a set ...
1
vote
1answer
45 views

Basis of the intersection of two spans

I am having troubles trying to solve this example: In $\mathbb{R}^4$, find a basis of $L1 \cap L2$, when $L_1=\operatorname{Span}\{a, b, c\}$ and $L_2=\operatorname{Span}\{d,e, f\}$ where: $a = (1, ...
0
votes
2answers
62 views

Determine which set span $\mathbb{R^3}$

Let $v_1,v_2,v_3$ be vectors in $\mathbb{R^3}$ such that $\langle v_1,v_2,v_3\rangle=\mathbb{R^3}$ Determine which of the following sets span $\mathbb{R^3}$ i)$S=\{v_1,v_2\}$ ...
3
votes
2answers
212 views

Is it possible that there isn't a linear span which precisely spans a vector space?

In the assignment I'm asked to decide whether given: $S = \Bigg \{ \begin{bmatrix}a &b \\ c &d\end{bmatrix} \in M_2(\mathbb{R}) \; | \; ad = 0 \Bigg \},\mathbb{F} = \mathbb{R}$. $S$ is a ...
0
votes
1answer
49 views

Find a minimal spanning set of a set of matrices

I'm supposed to find a minimal spanning set of $W = \{A \in M_n(\mathbb{R}) | \operatorname{Tr}(A) = 0\}$ First of all, what is a minimal spanning set? I can't find the term anywhere in the notes my ...
0
votes
1answer
25 views

Linear Algebra - Inner Products, Functions, and Closet Polynomial

This is the question: Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt ...
0
votes
1answer
40 views

Find bases for orthogonal complement $S^\perp$ for the subspace $S$

I'm having a tough time understanding the textbook on how to answer this question? I'm not too sure what to do? Any help will be appreciated. $$ S=\operatorname{span}\left[ \begin{pmatrix} 1 \\ -3 ...
0
votes
0answers
158 views

How to show a set of vectors does not span a vector space?

Let's say I am given a $4\times4$ matrix and I am to determine whether the columns of that matrix span $\mathbb R^4$. Please tell me if I'm correct: One way to determine that is to calculate the ...
1
vote
1answer
33 views

Find the orthonormal vectors $q_{1}, q_{2}, q_{3}$ such that $q_{1}, q_{2}$ span the column space of $A$?

We have given the matrix $$ A= \begin{pmatrix} 1 &1 \\ 2& -1 \\ -2 & 4 \end{pmatrix}$$ First the question asks find the orthonormal vectors $q_{1}, q_{2}, q_{3}$ such that $q_{1}, ...
0
votes
1answer
64 views

If $S\subset W$ and $W$ is subspace, is it ok to say $\operatorname{span}(S)\subseteq W$

I mean, if $S\subset W$ and $W$ is a subspace, then $S$ is either a basis for $W$ or at least spans some subset of $W$, therefore $\operatorname{span}(S)\subseteq W$. Is it ok? For finite sets it's ...
0
votes
1answer
17 views

Definition of onto for linear transformation

I had a question ask the following: "A linear transformation is onto if and only if the columns of its standard matrix form a generating set for its range." To me that seems true but the answer was ...
2
votes
0answers
86 views

$\rm span(S_1) + \rm span(S_2) = \rm span(S_1 \cup S_2)$ 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) ...