# linearly independent and dependent

• Mar 5th 2010, 10:17 PM
deniselim17
linearly independent and dependent
Problem 1:
If $\displaystyle u_{1}, u_{2},u_{3}$ are linearly independent vectors in $\displaystyle R^{n}$, show that $\displaystyle \{u_{1}, u_{3} \}$ is linearly independent set.

Proof 1:
If $\displaystyle a_{1}u_{1}+a_{3}u_{3}=0$ where $\displaystyle a_{1}, a_{3} \in R$, $\displaystyle a_{1}=a_{3}=0$ since $\displaystyle u_{1},u_{2},u_{3}$ is linearly independent.

Is the proof correct? I felt something missing in the proof.

Problem 2:
If $\displaystyle u_{1}, u_{2},u_{3}$ are linearly dependent vectors in $\displaystyle R^{n}$, show that $\displaystyle \{u, u_{1}, u_{2}, u_{3} \}$ is linearly dependent set for any $\displaystyle u \in R^{n}$.

Proof 2:
If $\displaystyle au+a_{1}u_{1}+a_{2}u_{2}+a_{3}u_{3}=0$. Since $\displaystyle u_{1}, u_{2}, u_{3}$ are linearly dependent, $\displaystyle a_{i} \neq 0$ for some $\displaystyle i=1,2,3$. Thus, $\displaystyle \{u, u_{1}, u_{2}, u_{3} \}$ is linearly dependent.

Please check my proofs.

Problem 3:
Let $\displaystyle u_{1}, ..., u_{k} \in R^{n}$ and $\displaystyle A \in M_{n}(R)$, $\displaystyle A$ is invertible .
Show that $\displaystyle u_{1}, ..., u_{k}$ are linearly independent if and only if $\displaystyle u_{1}A, ..., u_{k}A$ are linearly independent vectors in $\displaystyle R^{n}$.

How to prove problem 3? what theorem do I need?
• Mar 6th 2010, 01:58 AM
tonio
Quote:

Originally Posted by deniselim17
Problem 1:
If $\displaystyle u_{1}, u_{2},u_{3}$ are linearly independent vectors in $\displaystyle R^{n}$, show that $\displaystyle \{u_{1}, u_{3} \}$ is linearly independent set.

Proof 1:
If $\displaystyle a_{1}u_{1}+a_{3}u_{3}=0$ where $\displaystyle a_{1}, a_{3} \in R$, $\displaystyle a_{1}=a_{3}=0$ since $\displaystyle u_{1},u_{2},u_{3}$ is linearly independent.

Is the proof correct? I felt something missing in the proof.

Please check you haven't yet proved anything...(Wink) . You could argue as follows:

Suppose $\displaystyle 0=a_1u_1+a_3u_3=a_1u_1+0\cdot u_2+a_3u_3\Longrightarrow a_1=0=a_3$ since we're given $\displaystyle \{u_1,u_2,u_3\}$ is lin. ind.

Problem 2:
If $\displaystyle u_{1}, u_{2},u_{3}$ are linearly dependent vectors in $\displaystyle R^{n}$, show that $\displaystyle \{u, u_{1}, u_{2}, u_{3} \}$ is linearly dependent set for any $\displaystyle u \in R^{n}$.

Proof 2:
If $\displaystyle au+a_{1}u_{1}+a_{2}u_{2}+a_{3}u_{3}=0$. Since $\displaystyle u_{1}, u_{2}, u_{3}$ are linearly dependent, $\displaystyle a_{i} \neq 0$ for some $\displaystyle i=1,2,3$. Thus, $\displaystyle \{u, u_{1}, u_{2}, u_{3} \}$ is linearly dependent.

Please check my proofs.

This is similar as the above one but at least here you mentioned that the given set of vectors is l.i....I'd wrap it up as follows:

Since $\displaystyle \{u_{1}, u_{2},u_{3}\}$ is l.d. we have a trivial linear combination $\displaystyle a_1u_1+a_2u_2+a_3u_3=0$ and $\displaystyle a_i\neq 0$ for some $\displaystyle 1\leq i\leq 3\Longrightarrow 0\cdot u+a_1u_1+a_2u_2+a_3u_3=0$ and

STILL $\displaystyle a_i\neq 0$ for some index $\displaystyle i\,\Longrightarrow \{u,u_1,u_2,u_3\}$ is l.d.

Problem 3:
Let $\displaystyle u_{1}, ..., u_{k} \in R^{n}$ and $\displaystyle A \in M_{n}(R)$, $\displaystyle A$ is invertible .
Show that $\displaystyle u_{1}, ..., u_{k}$ are linearly independent if and only if $\displaystyle u_{1}A, ..., u_{k}A$ are linearly independent vectors in $\displaystyle R^{n}$.

How to prove problem 3? what theorem do I need?

This problem is sensibly more advanced than the other two, which just check you've actually understood the definition of l.d./l.i. sets ; here we need the lemma:

Lemma: A square matrix $\displaystyle n\times n\,\,A$ is invertible iff $\displaystyle \ker A=\{0\}\Longleftrightarrow vA=0\Longleftrightarrow v=0$

Try now to prove the question...

Tonio