**Problem 4: Linear Independence of Vectors** Verify whether the vectors \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\) are linearly independent. If yes, then find the determinant of the matrix with \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\) as rows. If not, then find linearly independent vectors that span the same linear space as \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\). **Question a.** \[ \vec{v}_1 = (1, 0, -1, 0), \] \[ \vec{v}_2 = (0, 1, 0, 1), \] \[ \vec{v}_3 = (1, 0, -1, 1), \] \[ \vec{v}_4 = (0, 1, 0, -1). \] **Question b.** \[ \vec{v}_1 = (1, 1, 0, 0), \] \[ \vec{v}_2 = (1, 0, -1, 0), \] \[ \vec{v}_3 = (0, 1, 0, -1), \] \[ \vec{v}_4 = (0, 0, 1, 1). \]

Problem 4: Linear Independence of Vectors Verify whether the vectors \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\) are linearly independent. If yes, then find the determinant of the matrix with \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\) as rows. If not, then find linearly independent vectors that span the same linear space as \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\). Question a. \[ \vec{v}_1 = (1, 0, -1, 0), \] \[ \vec{v}_2 = (0, 1, 0, 1), \] \[ \vec{v}_3 = (1, 0, -1, 1), \] \[ \vec{v}_4 = (0, 1, 0, -1). \] Question b. \[ \vec{v}_1 = (1, 1, 0, 0), \] \[ \vec{v}_2 = (1, 0, -1, 0), \] \[ \vec{v}_3 = (0, 1, 0, -1), \] \[ \vec{v}_4 = (0, 0, 1, 1). \]

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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Question

Verify whether vectors ~v1, ~v2, ~v3, ~v4 are linearly independent. If yes
then find the determinant of the matrix with ~v1, ~v2, ~v3, ~v4 as rows. If not
then find linearly independent vectors that span the same linear space as
~v1, ~v2, ~v3, ~v4.

Question a)

v1=(1, 0, −1, 0),v2=(0, 1, 0, 1),v3=(1, 0, −1, 1) v4=(0, 1, 0, −1)

Question b)

~v1=(1, 1, 0, 0) , v2 = (1, 0, −1, 0),v3 = (0, 1, 0, −1),v4 = (0, 0, 1, 1).

**Problem 4: Linear Independence of Vectors**

Verify whether the vectors \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\) are linearly independent. If yes, then find the determinant of the matrix with \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\) as rows. If not, then find linearly independent vectors that span the same linear space as \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4\).

**Question a.**

\[
\vec{v}_1 = (1, 0, -1, 0),
\]

\[
\vec{v}_2 = (0, 1, 0, 1),
\]

\[
\vec{v}_3 = (1, 0, -1, 1),
\]

\[
\vec{v}_4 = (0, 1, 0, -1).
\]

**Question b.**

\[
\vec{v}_1 = (1, 1, 0, 0),
\]

\[
\vec{v}_2 = (1, 0, -1, 0),
\]

\[
\vec{v}_3 = (0, 1, 0, -1),
\]

\[
\vec{v}_4 = (0, 0, 1, 1).
\]

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