Let ẞ = {v1, v2, ••• , vn} be a basis for a vector space V and let u1, ... , uk be vectors in V. Then {u1, ... , uk} is linearly independent in V if and only if { [ u1]ẞ , ... , [ uk ]ẞ} is linearly independent in Rn ,then prove that { u1, ... , uk} is linearly independent in V.

Answered: Let ẞ = {v1, v2, ••• , vn} be a basis…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let ẞ = {v₁, v₂, ••• , v_n} be a basis for a vector space V and let u₁, ... , u_k be vectors in V. Then {u₁, ... , u_k} is linearly independent in V if and only if { [ u₁]ẞ , ... , [ u_k ]ẞ} is linearly independent in Rn ,then prove that { u₁, ... , u_k} is linearly independent in V.

Expert Solution

Step 1

It is given that, $B = \{v_{1}, v_{2}, . . ., v_{n}\}$ be a basis for a vector space V.

And, $u_{1}, u_{2}, . . ., u_{k}$ be any vectors in V.

If $\{{[u_{1}]}_{B}, {[u_{2}]}_{B}, . . ., {[u_{k}]}_{B}\}$ is linearly independent in $ℝ^{n}$ .

Then, we have to show that $\{u_{1}, u_{2}, . . ., u_{k}\}$ is linearly independent in V.

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