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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1answer
39 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 ...
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1answer
18 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
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1answer
22 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 ...
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0answers
37 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
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1answer
26 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}, ...
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1answer
61 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 ...
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1answer
10 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 ...
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0answers
77 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) ...
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1answer
43 views

Span set of a matriecs

I am not sure if my answer is correct. If $$S=\left\{\begin{pmatrix} a &b \\ c & d \end{pmatrix}\;\middle\vert\; ad=0 \quad a,b,c,d\in \mathbb{R} \right\}$$ it means that or $a=0$ or $d=0$ ...
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1answer
50 views

Checking whether a vector is in the span of a set of vectors

Suppose we have a set of vectors $S = \{u,v,w\}$ in $\mathbb{R}^3$. We want to find if some vector $x$ is in the span of $S$. From what I understand, for $x$ to be in the span of $S$, we need to come ...
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0answers
11 views

Show that the subspace $B_1$ is a basis of $C^4$

I have $B_1$ = $((i,0,0,0),(1,0,1,i),(0,2,i,0),(-i,0,0,i))$ And $C^4$ is a vector space and a basis of it is $C_b$ = $(e_1,e_2,e_3,e_4)$ I want to show that $B_1$ is a basis of $C_b$. So i introduce ...
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0answers
33 views

A linear system for which the solution space is spanned by the given vectors [duplicate]

Make a system with $3$ equations and $3$ unknowns of which the solution space $V$ is spanned by the column vectors: $$\begin{bmatrix} 1 \\0 \\-1\end{bmatrix},\quad \begin{bmatrix}1 \\3\\ ...
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1answer
70 views

Prune a linearly independent set? What is the element of Span(Z)? What is the

Consider the following subset of $P_3(\mathbb{R})$ (real polynomial functions of degree at most 3). $$ Z = \{f_1, f_2, f_3, f_4, f_5\} $$ where $f_1(x) = 1-2x+2x^2-x3$, $f_2(x) = 1-x+x^2+x^3$, ...
3
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1answer
94 views

Linear system for which the solution space is spanned by the given vectors

Make a system with $3$ equations and $3$ unknowns of which the solution space $V$ is spanned by the column vectors: $$\begin{bmatrix} 1 \\0 \\-1\end{bmatrix},\quad \begin{bmatrix}1 \\3\\ ...
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0answers
29 views

Do prove in vector space (about span and subspace)

(a) let vector u = (a,b,b,0) and vector v = (0,c,-c,d) because u dot v = 0, thus v = c(0,1,-1,0) +d (0,0,0,1) therefore, W2 = span {(0,1,-1,0),(0,0,0,1)} the basis for w2 is (0,1,-1,0),(0,0,0,1) for ...
0
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1answer
20 views

Find the basis of set given by matrices

In linear space of matrix $2\times 3$ over $C$ we have subspace generated by: $ A= \{{\left[\begin{array}{ccc}i&i&i\\i&0&1\end{array}\right]}$ ...
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0answers
25 views

Find a basis for the span of each set?

I found the span of the set. Then I used GJ to get the RREF, and used the row reduced rows to form the basis. I got the basis as <( 1 0 -2 ; 0 1 1 )> However, my lecturer went a different way, ...
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1answer
29 views

show there exist non zero vector which is linear combination of other

sLet $a_1, \ldots , a_n$ be a basis of linear space $V$ let $W \le V$ be a $k$ dimensional subspace $k \ge 1$ Show for each subset $\displaystyle a_{i_i}, \ldots a_{i_m}$ for $m>n-k$ exist non ...
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3answers
43 views
0
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1answer
33 views

Basis for solution space?

For the matrix: $$ \begin{bmatrix} 1 & 0 & 2 & | & 0 \\ 0 & 1 & 3 & | & 0 \\ 0 & 0 & 0 & | & 0 \end{bmatrix} $$ ...
0
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1answer
26 views

Linear Algebra Spanning question

If I have two 3x1 column vectors in a vector space V that are linearly independent, how can I make a system of 3 eqns whose solution will span V? For example, column vector [1,3,0], column vector ...
1
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1answer
34 views

Find a matrix X∈V such that U∩W=span{X}

Here is my problem. I've tried reading other people's related questions, but they're always just slightly different, I can't find one like mine and don't really know how to approach this problem. ...
0
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1answer
14 views

Confused about the dimension of a span of a set of vectors ls

The question is: What is the dimension of the following subspace of $\mathbb{R^5}$? $$span\left( \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ -1 \\ 1 \\ 0 ...
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3answers
43 views

Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$

Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$. Ok I'm finally understanding what each of these ...
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0answers
33 views

understanding and visualizing the span of sets

I've been researching for a while and trying to wrap my head around spanning of vector spaces completely (by visualizing them in R3) before moving on to Linear Independence, Basis' and anything else ...
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2answers
40 views

Proving linearly independent vectors [closed]

Let $u$, $v$, $w$ be three linearly independent vectors in $\mathbb{R}^3$, and let $A$ be a non-singular $3\times3$ matrix. Then vectors $Au$, $Av$, $Aw$ are also linearly independent. This is a ...
2
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1answer
26 views

how to find out if something in general(scalar, function, vector) is in the span of a set

im pretty sure that if a vector is in the span of a set of vectors, then it can be written as a linear combination of the vectors in the set, whch you can find out by setting up a system of equations. ...
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2answers
32 views

Is $span(v_1, . . . ,v_m)$ a linearly dependent or linearly independent set of vectors? Also, what will happen if we take span of span?

We know that, the set of all linear combinations of $(v_1, . . . ,v_m)$ is called the span of $(v_1, . . . ,v_m)$, denoted by $span(v_1, . . . ,v_m)$. In other words, \begin{align} span(v_1, . . . ...
1
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1answer
26 views

Knowing if spans overlap

Only the first checked squares are deemed to be correct. Why is D not correct? After all, the vectors do overlap on the same plane...
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0answers
12 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 ...
0
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1answer
25 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 ...
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0answers
17 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
115 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
64 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, ...
4
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2answers
55 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
81 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
33 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
12 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, ...
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2answers
147 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 ...
1
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1answer
43 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
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2answers
24 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
13 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
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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
228 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
38 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 ...
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0answers
177 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 ...
0
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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 ...
2
votes
3answers
120 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 $3\times3$ ...
0
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2answers
47 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 $V = \mathrm{span}(v_1,v_2,v_3)$ ...