Question

EXERCISE 4. (i) Let W be the plane in $\mathbb{R}^3$ defined by the equation $4x + y - 3z = 0$. Find a basis $B$ for this subspace of $\mathbb{R}^3$. (ii) Show that the vector $v = (2, 1, 3)$ lies in $W$ and express it in terms of the basis $B$. (iii) If $(w)_B = (-2, 4)$ then what is $(w)_S$ (where $S$ is the standard basis of $\mathbb{R}^3$)?

          EXERCISE 4. (i) Let W be the plane in $\mathbb{R}^3$ defined by the equation
$4x + y - 3z = 0$.
Find a basis $B$ for this subspace of $\mathbb{R}^3$.
(ii) Show that the vector $v = (2, 1, 3)$ lies in $W$ and express it in terms of the basis $B$.
(iii) If $(w)_B = (-2, 4)$ then what is $(w)_S$ (where $S$ is the standard basis of $\mathbb{R}^3$)?

EXERCISE 4. (i) Let W be the plane in ℝ^3 defined by the equation
4x + y - 3z = 0.
Find a basis B for this subspace of ℝ^3.
(ii) Show that the vector v = (2, 1, 3) lies in W and express it in terms of the basis B.
(iii) If (w)B = (-2, 4) then what is (w)S (where S is the standard basis of ℝ^3)?

Added by Sara L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Ben Blakesley

Step 1

First, let's find one vector that lies in the plane. We can set z = 0 and solve for x and y: 4x + y - 3(0) = 0 4x + y = 0 y = -4x So one vector that lies in the plane is (1, -4, 0). Show more…

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Exercise 4.i: Let W be the plane in R^3 defined by the equation 4x + y - 3z = 0. Find a basis B for this subspace of R^3. Show that the vector v = (2, 1, 3) lies in W and express it in terms of the basis B. (ii) If w = -2, then what is w_s where S is the standard basis of R^3?