# Thread: linearly independent and dependent

1. ## 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.

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?

2. 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... . 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.

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