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## Homework Statement

I'm studying for my linear algebra midterm, one of the challenge questions from my textbook is as follows:

Using the procedure of Example 8 of Chapter 2.3, find whether or not {(0,1,0,1),(-1,1,4,1),(-1,0,2,2)} is or is not a basis for the hyperplane 4[itex]x_{1}[/itex]-[itex]x_{2}[/itex]+[itex]x_{3}[/itex]+[itex]x_{4}[/itex]=0 in [itex]ℝ^{n}[/itex]

__Example 8:__

Show that [itex]\beta[/itex]={[1,2,-1],[1,1,1]} is a basis for the plane -3[itex]x_{1}[/itex]+2[itex]x_{2}[/itex]+[itex]x_{3}[/itex]=0

We observe that [itex]\beta[/itex] is clearly linearly independent since neither vector is a scalar multiple of the other. Thus, we need to show that every vector in the plane can be written as a linear combination of the vectors in [itex]\beta[/itex]. To do this, observe that any vector [itex]\vec{X}[/itex] in the plane must satisfy the condition of the plane. Hence, every vector in the plane has the form

[itex]\vec{X}[/itex] = [([itex]x_{1}[/itex]),([itex]x_{2}[/itex]),(3[itex]x_{1}[/itex]-2[itex]x_{2}[/itex])]

Since [itex]x_{3}[/itex]=3[itex]x_{1}[/itex]-2[itex]x_{2}[/itex]

Therefore, we now just need to show that the equation

t1(1,2,-1)+t2(1,1,1)=[([itex]x_{1}[/itex]),([itex]x_{2}[/itex]),(3[itex]x_{1}[/itex]-2[itex]x_{2}[/itex])]

is always consistent

Row reducing the corresponding augmented matrix gives

[(1,0,0), (1,1,0)|(([itex]x_{1}[/itex]),(2[itex]x_{1}[/itex]-[itex]x_{2}[/itex]),(0))]

## Homework Equations

## The Attempt at a Solution

I'm not entirely sure where to start with this one. I've been working really hard in this class, but it's not sticking. Thank you