3. (15 points) Find a basis for the solution space of the following linear syste

Question

3. (15 points) Find a basis for the solution space of the following linear system of equa tions and explain why it is a basis. Determine the dimension of the solution space xi + 2x2 T2 T3 2x3 4 3x4 Solution: First we obtain x4-0, the we let x3-8, We can also express in vector form 3 2 X-S SV This shows that all solutions can be expressed in terms of the vector v - (-3,2,1,0) so the set containing v is the spanning set of the solution space, it is a linearly independent set with just one vector, so the dimension of the solution space is just 1

Explanation / Answer

Third equation gives us x4=0

Now we have two equations and three variables to be determined namely,x1,x2,x3

And the first two equations are not multiples of each other hence we have 1 free variable

So we choose, x3, to be free variable and determine,x1 and x2 in terms of x3

x1+2x2=x3

x2=2x3

x1+4x3=x3

x1=-3x3

v=(-3x3 2x3 x3 0)^T

v=x3(-3 2 1 0)^T

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