Using the given basis of a vector to prove other basis

In summary: Therefore, we have shown that $\left\{{f}_{1},{f}_{2},{f}_{3}\right\}$ is a basis for $V$ as well. In summary, the set $\left\{{f}_{1},{f}_{2},{f}_{3}\right\}$ is also a basis for vector space $V$ over the field $F$. This can be proven by showing that $e_1$, $e_2$, and $e_3$ are all part of the linear span of $f_1$, $f_2$, and $f_3$, and since they form a basis for $V$, this means that $f_1$, $f_2$, and $f
  • #1
Kronos1
5
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Hi Guys having a bit of trouble understanding vector basis.

If \(\displaystyle \left\{{e}_{1},{e}_{2},{e}_{3}\right\}\) is a basis for vector space $V$ over the field $F$

and \(\displaystyle {f}_{1}=-{e}_{1}, {f}_{2}={e}_{1}-{e}_{2}, {f}_{3}={e}_{1}-{e}_{3}\)

how can I go about proving that \(\displaystyle \left\{{f}_{1},{f}_{2},{f}_{3}\right\}\) is also a basis for $V$?
 
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  • #2
Kronos said:
Hi Guys having a bit of trouble understanding vector basis.

If \(\displaystyle \left\{{e}_{1},{e}_{2},{e}_{3}\right\}\) is a basis for vector space $V$ over the field $F$

and \(\displaystyle {f}_{1}=-{e}_{1}, {f}_{2}={e}_{1}-{e}_{2}, {f}_{3}={e}_{1}-{e}_{3}\)

how can I go about proving that \(\displaystyle \left\{{f}_{1},{f}_{2},{f}_{3}\right\}\) is also a basis for $V$?

Hi Kronos,

Solve the system of equations for $e_1$, $e_2$, and $e_3$. Your resulting equations will show that $e_1$, $e_2$, and $e_3$ all belong to the linear span of $f_1$, $f_2$, and $f_3$. Thus Span($e_1$, $e_2$, $e_3$) $\subset$ Span($f_1$, $f_2$, $f_3$). Since $e_1$, $e_2$, and $e_3$ form a basis for $V$, they span $V$ and thus $V = \text{Span}(f_1, f_2, f_3)$. This means that $f_1$, $f_2$, and $f_3$ span $V$. The $f$-vectors also linearly independent since $V$ is three-dimensional (as $e_1$, $e_2$, and $e_3$ form a basis for $V$).
 

FAQ: Using the given basis of a vector to prove other basis

How do you determine if a given set of vectors forms a basis?

To determine if a given set of vectors forms a basis, you can check if the vectors are linearly independent and span the vector space. This can be done by setting up a system of equations and solving for the coefficients of the linear combination. If the only solution is the trivial solution (all coefficients are zero), then the vectors are linearly independent. To check if the vectors span the vector space, you can substitute them into the basis vectors and see if they can be expressed as a linear combination of the basis vectors.

Can you use any set of vectors as a basis for a vector space?

No, not every set of vectors can be used as a basis for a vector space. The vectors must be linearly independent and span the vector space in order to form a basis. Additionally, the vector space must have a finite number of dimensions.

How do you prove that a set of vectors forms a basis using a given basis?

To prove that a set of vectors forms a basis using a given basis, you can use the concept of change of basis. This involves expressing the given basis vectors as a linear combination of the new basis vectors. If this is possible, then the new basis vectors span the same vector space as the given basis vectors, and are therefore a basis themselves.

Can you use a different number of vectors in a basis to prove a different number of vectors in another basis?

Yes, it is possible to use a different number of vectors in a basis to prove a different number of vectors in another basis. As long as the vectors are linearly independent and span the same vector space, they can be used as a basis.

Is it necessary to use a basis to prove other basis?

Yes, using a basis is necessary to prove other basis. A basis is a set of vectors that can be used to represent all other vectors in a vector space. Without a basis, it would be difficult to prove the existence of a set of vectors that can span the same vector space.

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