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I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for.

I understand that the Span of a Vector Space $V$ is the linear combination of all the vectors in $V$.

I also understand that the Basis of a Vector Space V is a set of vectors ${v_{1}, v_{2}, ..., v_{n}}$ which is linearly independent and whose span is all of $V$.

Now, from my understanding the basis is a combination of vectors which are linearly independent, for example, $(1,0)$ and $(0,1)$.

But why?

The other question I have is, what do they mean by "whose span is all of $V$" ?

On a final note, I would really appreciate a good definition of Span and Basis along with a concrete example of each which will really help to reinforce my understanding.

Thanks.

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    $\begingroup$ Here is a lecture from Gilbert Strang (MIT) about Span and Basis. His explanation is very good, so posting the video here instead of instead of answering it again. All the sessions around Linear Algebra can be found here. $\endgroup$ – Praveen Sripati Sep 8 '16 at 10:18
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Span is usually used for a set of vectors. The span of a set of vectors is the set of all linear combinations of these vectors.

So the span of $\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ would be the set of all linear combinations of them, which is $\mathbb{R}^2$. The span of $\{\begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ is also $\mathbb{R}^2$, although we don't need $\begin{pmatrix}2\\0\end{pmatrix}$ to be so.

So both these two sets are said to be the spanning sets of $\mathbb{R}^2$.

However, only the first set $\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ is a basis of $\mathbb{R}^2$, because the $\begin{pmatrix}2\\0\end{pmatrix}$ makes the second set linearly dependent.

Also, the set $\{\begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ can also be a basis for $\mathbb{R}^2$. Because its span is also $\mathbb{R}^2$ and it is linearly independent.

For another example, the span of the set $\{\begin{pmatrix}1\\1\end{pmatrix}\}$ is the set of all vectors in the form of $\begin{pmatrix}a\\a\end{pmatrix}$.

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  • $\begingroup$ So a basis of $\mathbb R^n$ is a smallest spanning set of $\mathbb R^n$, i.e any spanning set of $\mathbb R^n$ with cardinality $n$? $\endgroup$ – Zaz Oct 26 '18 at 23:17
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    $\begingroup$ @Zaz: That is correct. $\endgroup$ – KittyL Oct 30 '18 at 10:28
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The basis is a combination of vectors which are linearly independent and which spans the whole vector V.

Suppose we take a system of $R^2$. Now as you said,$(1,0)$ and $(0,1)$ are the basis in this system and we want to find any $(x,y)$ in this system.

$$ (x,y)=x(1,0)+y(0,1) $$ where $x$ , $y$ belong to a set of real numbers.

Using the linear combination of of the basis , you can find find any vector in the system.

For instance, $$ (98745,12345)=98745(1,0)+12345(0,1) $$

Now , what is span?

Span is the set of all linear combination vectors in the system.

In $R^2$,suppose span is the set of all combinations of $(1,0)$ and $(0,1)$.

This set would contain all the vectors lying in $R^2$,so we say it contains all of vector V.

Therefore, Basis of a Vector Space V is a set of vectors $v_1,v_2,...,v_n$ which is linearly independent and whose span is all of V.

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The span of a finite subset $S$ of a vector space $V$ is the smallest subvector space that contains all vectors in $S$. One shows easily it is the set of all linear combinations of lelements of $S$ with coefficients in the base field (usually $\mathbf R,\mathbf C$ or a finite field).

A basis of the vector space $V$ is a subset of linearly independent vectors that span the whole of $V$. If $S=\{x_1, \dots, x_n\}$ this means that for any vector $u\in V$, there exists a unique system of coefficients such that $$u=\lambda_1 x_1+\dots+\lambda_n x_n. $$

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