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Sean $U = \{(x, y, z) \in \mathbb{R}^3 : 3x - 2y + z = 0\}$ y $W = \{(x, y, z) \in \mathbb{R}^3 : -x + 2y + 2z = 0\}$ . Encontrar $U \cap W$ y calcular $\text{dim}(U \cap W)$ .

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Transformations of functions

Full solution

Q. Sean

U = \{(x, y, z) \in \mathbb{R}^3 : 3x - 2y + z = 0\}

y

W = \{(x, y, z) \in \mathbb{R}^3 : -x + 2y + 2z = 0\}

. Encontrar

U \cap W

y calcular

\text{dim}(U \cap W)

.

Identify Equations for U and W: Identify the equations for U and W. $\newline$ U is defined by $3x - 2y + z = 0$ and W is defined by $-x + 2y + 2z = 0$ .
Set Up System of Equations: Set up the system of equations to find $U \cap W$ . We solve: $1.$ $3x - 2y + z = 0$ $2.$ $-x + 2y + 2z = 0$
Solve Using Substitution or Elimination: Solve the system using substitution or elimination. $\newline$ Multiply the second equation by $3$ to align coefficients for elimination: $\newline$ $3(-x + 2y + 2z) = 0 \rightarrow -3x + 6y + 6z = 0$ $\newline$ Now add this to the first equation: $\newline$ $3x - 2y + z + (-3x + 6y + 6z) = 0$ $\newline$ $0x + 4y + 7z = 0$
Solve for One Variable: Solve for one variable in terms of others. $\newline$ From $0x + 4y + 7z = 0$ , express $y$ in terms of $z$ : $\newline$ $y = -\frac{7z}{4}$
Substitute Back to Find $x$ : Substitute $y$ back into one of the original equations to find $x$ . $\newline$ Using $3x - 2(-7z/4) + z = 0$ : $\newline$ $3x + 7z/2 + z = 0$ $\newline$ $3x + 9z/2 = 0$ $\newline$ $x = -3z/2$
Write General Solution: Write the general solution for the intersection $U \cap W$ . The general solution is $(x, y, z) = (-\frac{3z}{2}, -\frac{7z}{4}, z)$ , where $z$ is any real number. This can be rewritten parametrically as: $(x, y, z) = z(-\frac{3}{2}, -\frac{7}{4}, 1)$
Determine Dimension of $U \cap W$ : Determine the dimension of $U \cap W$ . Since the solution depends on one parameter ( $z$ ), $U \cap W$ is a $1$ -dimensional subspace of $\mathbb{R}^3$ .

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