For the linear system in Example 5 of Section 1.2, express the general solution that we obtained in that example as a linear combination of column vectors that contain only numerical entries. [ Suggestion : Rewrite the general solution as a single column vector, then write that column vector as a sum of column vectors each of which contains at most one parameter, and then factor out the parameters.] EXAMPLE 5 | Gauss–Jordan Elimination Solve by Gauss–Jordan elimination. x 1 + 3 x 2 − 2 x 3 + 2 x 5 = 0 2 x 1 + 6 x 2 − 5 x 3 − 2 x 4 + 4 x 5 − 3 x 6 = − 1 5 x 3 + 10 x 4 + 15 x 6 = 5 2 x 1 + 6 x 2 + 8 x 4 + 4 x 5 + 18 x 6 = 6
For the linear system in Example 5 of Section 1.2, express the general solution that we obtained in that example as a linear combination of column vectors that contain only numerical entries. [ Suggestion : Rewrite the general solution as a single column vector, then write that column vector as a sum of column vectors each of which contains at most one parameter, and then factor out the parameters.] EXAMPLE 5 | Gauss–Jordan Elimination Solve by Gauss–Jordan elimination. x 1 + 3 x 2 − 2 x 3 + 2 x 5 = 0 2 x 1 + 6 x 2 − 5 x 3 − 2 x 4 + 4 x 5 − 3 x 6 = − 1 5 x 3 + 10 x 4 + 15 x 6 = 5 2 x 1 + 6 x 2 + 8 x 4 + 4 x 5 + 18 x 6 = 6
For the linear system in Example 5 of Section 1.2, express the general solution that we obtained in that example as a linear combination of column vectors that contain only numerical entries. [Suggestion: Rewrite the general solution as a single column vector, then write that column vector as a sum of column vectors each of which contains at most one parameter, and then factor out the parameters.]
EXAMPLE 5 | Gauss–Jordan Elimination
Solve by Gauss–Jordan elimination.
x
1
+
3
x
2
−
2
x
3
+
2
x
5
=
0
2
x
1
+
6
x
2
−
5
x
3
−
2
x
4
+
4
x
5
−
3
x
6
=
−
1
5
x
3
+
10
x
4
+
15
x
6
=
5
2
x
1
+
6
x
2
+
8
x
4
+
4
x
5
+
18
x
6
=
6
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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