Let ( T: mathbb{R}^3 rightarrow mathbb{R}^3 ) be a linear transformation defined by ( T(x, y, z) = (2x + y - 3z, x - y + z, 3x + 2y - 4z) ). Find the kernel of ( T ).

Question

Accepted Answer

Step:1 ：Step 1: Set up the equation ( T(x, y, z) = 0 ), which gives us the system of equations: begin{align*} 2x + y - 3z &= 0  x - y + z &= 0  3x + 2y - 4z &= 0  end{align*}Step:2 ：Step 2: Solve the system of equations. We can use the method of substitution or elimination. By adding the first two equations, we get ( 3x = 2z ), or ( x = frac{2}{3}z ). Substituting this into the third equation, we get ( y = - frac{1}{3}z ).Step:3 ：Step 3: The solution set of the system of equations is the kernel of the transformation ( T ), so we obtain ( text{Ker}(T) = left{ (x, y, z) in mathbb{R}^3 : x = frac{2}{3}z, y = - frac{1}{3}z right} ).

Answer

Step: 1 ：Step 1: Set up the equation ( T(x, y, z) = 0 ), which gives us the system of equations: begin{align*} 2x + y - 3z &= 0  x - y + z &= 0  3x + 2y - 4z &= 0  end{align*}Step: 2 ：Step 2: Solve the system of equations. We can use the method of substitution or elimination. By adding the first two equations, we get ( 3x = 2z ), or ( x = frac{2}{3}z ). Substituting this into the third equation, we get ( y = - frac{1}{3}z ).Step: 3 ：Step 3: The solution set of the system of equations is the kernel of the transformation ( T ), so we obtain ( text{Ker}(T) = left{ (x, y, z) in mathbb{R}^3 : x = frac{2}{3}z, y = - frac{1}{3}z right} ).

Let $T : R^{3} \to R^{3}$ be a linear transformation defined by $T (x, y, z) = (2 x + y - 3 z, x - y + z, 3 x + 2 y - 4 z)$ . Find the kernel of $T$ .

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Popular Problems

Let T:R3→R3 be a linear transformation defined by T(x,y,z)=(2x+y−3z,x−y+z,3x+2y−4z). Find the kernel of T.

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Popular Problems

Let $T : R^{3} \to R^{3}$ be a linear transformation defined by $T (x, y, z) = (2 x + y - 3 z, x - y + z, 3 x + 2 y - 4 z)$ . Find the kernel of $T$ .