High School

Rewrite the general solution below in parametric vector form and list the basis vectors.

\[ x_2 = 9x_3 + 7 - 6x_5 + x_6, \]
\[ x_4 = 8 - 5x_6 + 2x_7, \]
where \( x_1, x_3, x_5, x_6, x_7 \) are free variables.

Answer :

The general solution with free variables x1, x3, x5, x6, and x7 is rewritten in parametric vector form, assigning parameters t1 to t5 for each free variable.

The solution then is a linear combination of parameters and basis vectors of the vector space.

The given general solution of a linear system, where x1, x3, x5, x6, and x7 are free variables, can be rewritten in parametric vector form. To express the solution set parametrically, we assign a parameter to each free variable:

  • Let t1 = x1
  • Let t2 = x3
  • Let t3 = x5
  • Let t4 = x6
  • Let t5 = x7

Using these parameters, the general solution can be written as:

x = x1e1 + (9t2+7-6t3+t4)e2 + t2e3 + (8-5t4+2t5)e4 + t3e5 + t4e6 + t5e7

Where ei are the basis vectors of the vector space.

The basis vectors corresponding to the free variables would be the unit vectors in the direction of each free variable axis:

  • e1 = (1, 0, 0, 0, 0, 0, 0)
  • e3 = (0, 0, 1, 0, 0, 0, 0)
  • e5 = (0, 0, 0, 0, 1, 0, 0)
  • e6 = (0, 0, 0, 0, 0, 1, 0)
  • e7 = (0, 0, 0, 0, 0, 0, 1)

And the non-free variables would get the following basis vectors:

  • e2 = (0, 1, 0, 0, 0, 0, 0)
  • e4 = (0, 0, 0, 1, 0, 0, 0)

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