Math Problem Statement

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 .

Solution

To determine a basis for the 2-dimensional subspace VV of R4\mathbb{R}^4 given by the following pair of equations in the coordinates [x,y,z,u][x,y,z,u]:

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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Vector Spaces

Formulas

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Theorems

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Suitable Grade Level

University