The 2
-dimensional subspace V
of R4
is given by means of the following pair of equations in the coordinates [x,y,z,u]
:
{20⋅y−10⋅z+10⋅u10⋅x−4⋅y+4⋅z==00

Determine a basis of V
in the form {a⃗ ,b⃗ }
in which a⃗
and b⃗
are row vectors of R4
.

To determine a basis for the 2-dimensional subspace \( V \) of \( \mathbb{R}^4 \) given by the following pair of equations in the coordinates \([x,y,z,u]\):

\[
\begin{aligned}
20y - 10z + 10u &= 0, \\
10x - 4y + 4z &= 0,
\end{aligned}
\]

we need to solve these equations and express the general solution in parametric form. This will allow us to extract the basis vectors.

### Step 1: Solve the system of equations

Given equations:

1. \( 20y - 10z + 10u = 0 \)
2. \( 10x - 4y + 4z = 0 \)

**Simplify the equations:**

1. Divide the first equation by 10:  
   \( 2y - z + u = 0 \)  
   \( u = z - 2y \)

2. Divide the second equation by 2:  
   \( 5x - 2y + 2z = 0 \)  
   \( 5x = 2y - 2z \)  
   \( x = \frac{2}{5}(y - z) \)

### Step 2: Express the general solution
We can express \( x \) and \( u \) in terms of the free variables \( y \) and \( z \):

\[
\begin{aligned}
x &= \frac{2}{5}(y - z), \\
y &= y, \\
z &= z, \\
u &= z - 2y.
\end{aligned}
\]

Thus, the general solution in vector form is:

\[
\begin{bmatrix} 
x \\
y \\
z \\
u 
\end{bmatrix} 
= y \begin{bmatrix} 
\frac{2}{5} \\
1 \\
0 \\
-2 
\end{bmatrix} 
+ z \begin{bmatrix} 
-\frac{2}{5} \\
0 \\
1 \\
1 
\end{bmatrix}.
\]

### Step 3: Identify the basis vectors

The basis vectors \( \vec{a} \) and \( \vec{b} \) are the coefficients of the free variables \( y \) and \( z \) respectively:

\[
\vec{a} = \begin{bmatrix} 
\frac{2}{5} \\
1 \\
0 \\
-2 
\end{bmatrix}, 
\quad
\vec{b} = \begin{bmatrix} 
-\frac{2}{5} \\
0 \\
1 \\
1 
\end{bmatrix}.
\]

### Final Answer
The basis for the 2-dimensional subspace \( V \) is:

\[
\left\{
\begin{bmatrix} 
\frac{2}{5} \\
1 \\
0 \\
-2 
\end{bmatrix}, 
\begin{bmatrix} 
-\frac{2}{5} \\
0 \\
1 \\
1 
\end{bmatrix}
\right\}.
\]

Would you like a detailed explanation of any particular step?

### Relative Questions:
1. How would the process change if the equations involved more variables?
2. Can you show how to determine if a set of vectors forms a basis for a subspace?
3. How can the rank of a matrix be used to find the dimension of a subspace?
4. How can you determine if a given vector lies within a subspace?
5. Can you provide examples of how subspaces apply to real-world problems?

### Tip:
When solving systems of linear equations, always look for ways to simplify the equations first. This can make finding solutions and interpreting the results much easier.

Explore how to determine a basis for a 2-dimensional subspace in R^4 defined by a system of linear equations. Understand the process of solving the equations and extracting basis vectors. Suitable for university-level studies in linear algebra and vector spaces.

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