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
-Subset

The Attempt at a Solution

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

Let ,n=3\Rightarrow [V]= \begin{bmatrix}<br /> a &amp; b &amp; b \\<br /> b &amp; a &amp; b \\<br /> b &amp; b &amp; a \\<br /> \end{bmatrix}

rref[V]=\begin{bmatrix}<br /> 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 \\<br /> \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

[/B]
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}<br /> a &amp; b &amp; b \\<br /> b &amp; a &amp; b \\<br /> b &amp; b &amp; a \\<br /> \end{bmatrix}

rref[V]=\begin{bmatrix}<br /> 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 \\<br /> \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

[/B]
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.