Is span a subset in ##\mathbb{R}^{n}##?

[yango_17]

Homework Statement

Consider the vectors $\vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}}$ in $\mathbb{R}^{n}$. Is span $(\vec{v_{1}},...,\vec{v_{m}})$ necessarily a subspace of $\mathbb{R}^{n}$? Justify your answer.

Homework Equations

The Attempt at a Solution

I understand the three conditions required for a subset to be a subspace (includes zero vector, closed under addition, closed under scalar multiplication), but I am not sure how to go about testing these properties with the span. Any help would be appreciated. Thanks.

 

[Mark44]

Homework Statement

Consider the vectors $\vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}}$ in $\mathbb{R}^{n}$. Is span $(\vec{v_{1}},...,\vec{v_{m}})$ necessarily a subspace of $\mathbb{R}^{n}$? Justify your answer.

Homework Equations

The Attempt at a Solution

I understand the three conditions required for a subset to be a subspace (includes zero vector, closed under addition, closed under scalar multiplication), but I am not sure how to go about testing these properties with the span. Any help would be appreciated. Thanks.

What's another way to write $span(\vec{v_{1}},...,\vec{v_{m}})$? How do you know whether a given vector is a member of this set?

 

[yango_17]

You can rewrite span as the image of a matrix, since the image of a matrix is the span of its columns. Since image is a subspace, then does it follow that span is a subspace?

 

[Mark44]

You can rewrite span as the image of a matrix

There's no need at all to use matrices. How does your book define the term "span"?

, since the image of a matrix is the span of its columns. Since image is a subspace, then does it follow that span is a subspace?

 

[yango_17]

Span: Consider the vectors $\vec{v_{1}},...,\vec{v_{m}}$ in $\mathbb{R}^{n}$. The set of all linear combinations $c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}$ of the vectors $\vec{v_{1}},...,\vec{v_{m}}$ is called their span:
$span(\vec{v_{1}},...,\vec{v_{m}})=\left \{ c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}:c_{1},...,c_{m} \right \}$

 

[Ray Vickson]

Span: Consider the vectors $\vec{v_{1}},...,\vec{v_{m}}$ in $\mathbb{R}^{n}$. The set of all linear combinations $c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}$ of the vectors $\vec{v_{1}},...,\vec{v_{m}}$ is called their span:
$span(\vec{v_{1}},...,\vec{v_{m}})=\left \{ c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}:c_{1},...,c_{m} \right \}$

OK, so, if $\vec{w}_1$ and $\vec{w}_2$ are in the span, is $\vec{w}_1 + \vec{w}_2$ also in the span? If $c$ is a constant, is $c \, \vec{w}_1$ in the span? Is the vector $\vec{0}$ in the span?

 

[Mark44]

Span: Consider the vectors $\vec{v_{1}},...,\vec{v_{m}}$ in $\mathbb{R}^{n}$. The set of all linear combinations $c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}$ of the vectors $\vec{v_{1}},...,\vec{v_{m}}$ is called their span:
$span(\vec{v_{1}},...,\vec{v_{m}})=\left \{ c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}:c_{1},...,c_{m} \right \}$

Presumably, you mean this:
$span(\vec{v_{1}},...,\vec{v_{m}})=\left \{ c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}:c_{1},...,c_{m} \in \mathbb{R} \right \}$