Linear algebra: Find the span of a set

[gruba]

Homework Statement

Find the span of $U=\{2,\cos x,\sin x:x\in\mathbb{R}\}$ ($U$ is the subset of a space of real functions) and $V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}$

Homework Equations

- Span
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The Attempt at a Solution

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Objects in $U$ :$2,\cos x,\sin x$ are linearly independent, so they span $\mathbb{R^3}$.

Let ,$n=3\Rightarrow [V]= \begin{bmatrix} a & b & b \\ b & a & b \\ b & b & a \\ \end{bmatrix}$

$rref[V]=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\Rightarrow$ vectors in $V$ span $\mathbb{R^3}$, if $a,b\neq 0$.

But because $V\subset\mathbb{R^n}\Rightarrow$ vectors span $\mathbb{R^{n-1}}$.

Is this correct?

 

[Mark44]

Homework Statement

Find the span of $U=\{2,\cos x,\sin x:x\in\mathbb{R}\}$ ($U$ is the subset of a space of real functions) and $V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}$

Homework Equations

- Span
-Subset

The Attempt at a Solution

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Objects in $U$ :$2,\cos x,\sin x$ are linearly independent, so they span $\mathbb{R^3}$.

Yes.

Let ,$n=3\Rightarrow [V]= \begin{bmatrix} a & b & b \\ b & a & b \\ b & b & a \\ \end{bmatrix}$

$rref[V]=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\Rightarrow$ vectors in $V$ span $\mathbb{R^3}$, if $a,b\neq 0$.

I don't know how valid your argument is, here. It's given that $V \subset \mathbb{R}^n$. Can you extend this to a statement about an n-dimensional space instead of a 3-dimensional space?

But because $V\subset\mathbb{R^n}\Rightarrow$ vectors span $\mathbb{R^{n-1}}$.

Is this correct?

Why do you conclude that the vectors span $\mathbb{R}^{n - 1}$?

 

[LCKurtz]

Homework Statement

Find the span of $U=\{2,\cos x,\sin x:x\in\mathbb{R}\}$ ($U$ is the subset of a space of real functions) and $V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}$

Homework Equations

- Span
-Subset

The Attempt at a Solution

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Objects in $U$ :$2,\cos x,\sin x$ are linearly independent, so they span $\mathbb{R^3}$.

They aren't even in $\mathbb{R}^3$; they belong to a space of real functions.

 

[Mark44]

To clarify/correct my "Yes" in post #2, the functions span a three-dimensional space of functions, not $\mathbb{R}^3$.

 

[nuuskur]

Objects in $U$ :$2,\cos x,\sin x$ are linearly independent, so they span $\mathbb{R^3}$.

Yes, these vectors are linearly independent.
No, these vectors span a space isomorphic to $\mathbb{R}^3$

Regarding $V$. If a=b, then all of those vectors are linearly dependent. If a=0, then the vectors will span a null space. If $a\neq 0$, then the system of vectors will be reduced to only one vector and so it would span $\mathbb{R}^1\subset \mathbb{R}^n$
If $a\neq b$, then all of the vectors are in fact linearly independent: one can construct an n x n matrix of said vectors and reduce it to a diagonal matrix.