Question

2. (50%) Let $u_1, u_2, \dots, u_k \in \mathbb{R}^n$ and $c \in \mathbb{R}$. Show that {$u_1, u_2, \dots, u_k$} is linearly independent if and only if {$u_1 + cu_2, u_2, \dots, u_k$} is linearly independent.

          2. (50%) Let $u_1, u_2, \dots, u_k \in \mathbb{R}^n$ and $c \in \mathbb{R}$. Show that {$u_1, u_2, \dots, u_k$} is linearly independent if and only if {$u_1 + cu_2, u_2, \dots, u_k$} is linearly independent.

$2. (50%) Let $u1, u2, \dots, uk \in \mathbb{R}^n$ and $c \in \mathbb{R}$. Show that {$u1, u2, \dots, uk$} is linearly independent if and only if {$u1 + cu2, u2, \dots, uk$} is linearly independent.$

Added by Sharon N.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Dominique Jan Tan

Step 1

.., un} is linearly independent. This means that the only solution to the equation a1u1 + a2u2 + ... + anun = 0 is a1 = a2 = ... = an = 0. Now, let's consider the set {u1 + cu2, u2, ..., un}. We want to show that this set is linearly independent. So, let's assume Show more…

Show all steps

Thanks for your feedback!

Let u1, u2, ..., un âˆˆ R^n and c âˆˆ R. Show that {u1, u2, ..., un} is linearly independent if and only if {u1 + cu2, u2, ..., un} is linearly independent.