# Tagged Questions

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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### Finding a basis of $\mathbb R^{4}$ containing specific vectors. How can different standard basis vectors can be added, where both result in a basis?

An exercise from my textbook asks me to find a basis of $\mathbb R^{4}$ containing $S = (u,v)$, where $u = (0,1,2,3), v = (2,-1,0,1)$. The method they describe involves adding vectors from the ...
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### Completing a basis for $\mathbb{R}^4$

Find a basis for $\mathbb{R}^4$ containing the vectors $(0, 1, 2, 3)$ and $(0, 1, 0, 1)$. Having trouble with this; I know that there are 4 vectors in the basis, and that any coordinate in ...
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### Find a matrix X∈V such that U∩W=span{X}.

Have been trying this problem for 4 hours still can't figure it out.
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### First Year Linear Algebra Proving Subspace and Basis of a Finite Support Function

I am not that familiar with formatting on here, so I placed the question here: I hope that is okay. Here is my attempt at the question. a) We need to show that W is closed under addition and ...
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### The column space of $A^2$ is all of $\mathbb R^n$ if and only if the column space of $A$ is all of $\mathbb R^n$

How would I go about proving the following statement? "Let $A$ be an $n \times n$ matrix. $\operatorname{Col}(A^2)=\mathbb{R}^n$ if and only if $\operatorname{Col}(A)=\mathbb{R}^n$" I started off by ...
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### Smallest subspace containing a set of vectors is the span of those vectors.

Show that $$span (v_1, ..., v_k)$$ is a subspace of $$R^n$$ and is the smallest subspace containing $$v_1, ..., v_k$$. I know if we assume $$v_1, ..., v_k$$ is an element of V where V is a subspace ...
1answer
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### Linear subspace spanned by a set of matrices

Let $A _{t} = \left[\begin{matrix}2 & t \\ 0 & 2\end{matrix}\right]$. Are the elements $I, A _{t}, A^2_{t}, A^3_{t}$ linearly independent in the set of matrices of type $2 \times 2$? What ...
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### Finding the span of 3 vectors

Let $V = \mathbb R^3$, a vector space over the reals. Find the span $W$ of $\{(1, 2, 1), (3, −1, −4), (0, 7, 7)\}$ in the form $\{(x, y, z) ∈ V \mid ax + by + cz = 0\}$ for some $a, b, c$. Find a ...
1answer
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### In each case determine if x lies in U = span{y, z}. If x is in U, write it as a linear combination of y and z; if x is not in U, show why not. [closed]

Determine if x lies in U = span{y, z}. If x is in U, write it as a linear combination of y and z; if x is not in U, show why not. $x = (2, −1, 0, 1)$, $y = (1, 0, 0, 1)$, and $z = (0, 1, 0, 1)$. I ...
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### determine if the given vectors span $\mathbb{R}^4$

Determine if the given vectors span $\mathbb{R}^4$ ${(1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), (0, 0, 0, 1)}$. I'm completely confused on this question. My textbook gives a different problem but in ...
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### Prove that if $\{u_1 ,\dots , u_k, v\}$ is linearly dependent then $v\in\text{span}\{u_1 ,\dots , u_k\}$

Let F be a field. Suppose $\{u_1 ,\dots , u_k\}\subseteq F^n$ is a linearly independent set, and $v \in F^n$ does not belong to this set. Prove that if $\{u_1 ,\dots , u_k, v\}$ is linearly dependent ...
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### Finding a linearly independent subset with the same span as a given linearly dependent set of vectors.

Given the following linearly dependent set of vectors in $\mathbb{R^3}$: $$\{(1,1,1)^T, (2,3,1)^T, (4,5,3)^T, (1,2,0)^T\}$$ how would I find a linearly independent subset with the same span? My ...
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### Dimensions of a linearly dependent set of vectors

If you have a set of 3 vectors which span a subspace, but the vectors are linearly dependent and hence the basis consists of 2 vectors, is the dimension of the SPAN 2 or 3?
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### How to find the span of a set of polynomials [closed]

how do I find the span of a set of polynomials? Specifically: S=(1, x-3, x^2 +2x)
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### how to find the span of 3 vectors

how do I find the span of 3 vectors: specifically for $(1, 1, 2)$; $(0, -1, 1)$;$(2, 5, 1)$ ? I know the answer is $(a, b, 3a-b)$. I just don't know how you get to that answer. Thanks in advance
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### How to turn span into linear equality constraint?

Suppose $$\text{Im}(A) = \text{span}\{\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ 0 \\ 1\\ 0 \end{bmatrix} \}$$ How would you go about turning this very set into a linear ...
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### Given special matrix, how to partition into linear independence and linear dependent set

Suppose we are given a matrix $B \in \mathbb{R}^{n\times n}$, and $A \in \mathbb{R}^n$ Then form a matrix $M$ $M = \begin{bmatrix} A & BA & B^2A & \ldots & B^n A \end{bmatrix}$ How ...
1answer
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### Show that the set $\{v_1, v_2, … , v_{k-1}\}$ cannot be a basis for V .

Let $\{v_1, v_2, ... , v_k\}$ be a linearly independent set of vectors in a vector space V . Show that the set $\{v_1, v_2, ... , v_{k-1}\}$ cannot be a basis for V . I am trying to prove this by ...
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### Do the columns of the given matrix span $\mathbb{R}^3$?

I was given the matrix: $$A=\left(\begin{matrix}-4 & -7 & 1 & 2\\ 0 & 0 & 3 & 8\\5 & -1 &1 & -4\end{matrix}\right)$$ I arranged them into an augmented matrix and ...
1answer
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### Spanning Set of Null Space of Matrix

I'm having a tough time finding any resources online or even in my book that give me a good way to solve problems like the one I have. I have a couple like it, so if someone could explain generally ...
1answer
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### Finding the Span of R3, Given a Set of Vectors

I'm studying for a midterm and one of the practice problems was: Inside R3, consider the vectors: v1 = (0, 0, 0), v2 = (1, -1, 0), v3 = (1, 1, 0), v4 = (1, 0, -1), v5 = (0, 1, 1), v6 = (0, 1, -1). ...
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### What is a Spanning Set of vectors? Why do we need Spanning Sets? [closed]

What is a Spanning Set of vectors? What is the use of Spanning Sets in the real world? Please, explain without using format mathematical notations.
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### Basis and dimensions

How do i find the a basis and dimension for $A[x]$? Consider the subset of $R[x]$ given by $A[x]:=\{q(x)$ element of $\mathbb R_4[x]$ such that $q(2)=0=q(-3)\}$ I'm a bit confused because there are ...
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### Linear Algebra: Matrix vs Span help.

The concept of span and matrix are getting jumbled, and I would like to get some clarification before I fall too far off the cliff. Suppose I have a 2x2 matrix (all rows/columns are independent). ...
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### Is vector $w$ in the span of $\{v_1,v_2,v_3\}$?

Let $v_1 = (1,0,-1)^T$, $v_2 = (2, 1, 3)^T$, $v_3 = (4,2,6)^T$ and $w = (7,4,7)^T$ From what I understand, if I set up an augment matrix in the form \begin{pmatrix}1&2&4&7\\ ...
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### Line, Plane or Hyperplane?

Does span=(2,-1,1,2), (-2,1,-1,-2) represent a line, plane or hyperplane in R4? We haven't learned matrices yet either
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### If X and Y are two sets of vectors in a vector space V, and if X $\subset$ Y, then is span X $\subset$ span Y?

If X and Y are two sets of vectors in a vector space V, and if X $\subset$ Y, then is span X $\subset$ span Y? If so, why is or isn't the span of X a subset of the span of Y? EDIT: Thank you for the ...
1answer
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### Linear span of general set in topological linear spaces

I'm studying Functional Analysis and I'm in doubt with the definition of linear span. The book states that: Let $\mathscr{L}$ be a topological linear space and let $\mathscr{M}$ be a linear ...
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### Equality of span of leading eigenvectors

Question: Assume $A$ is a full rank real $n\times n$ matrix and $B$ is a real symmetric, positive definite matrix of size $n\times n$. What conditions on $A$ ensures that the span of the first $p$ ...
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