Question

Let w be the subset of r3 consisting of all points on a line that passes through the origin. Such a line can be represented by the parametric equations x = at, y = bt, and z = ct. Use these equations to show that w is a subspace of r3?

Answer 1

To demonstrate that w is a subspace of R3, we verify that it contains the zero vector, is closed under vector addition, and closed under scalar multiplication. All conditions are satisfied using the parametric line equations x = at, y = bt, and z = ct.

Step-by-step explanation:

To show that w is a subspace of ℝ3, we must demonstrate that it satisfies three criteria: it must contain the zero vector, it must be closed under vector addition, and it must be closed under scalar multiplication.

Firstly, when t = 0, the parametric equations x = at, y = bt, and z = ct all yield 0, providing the zero vector (0, 0, 0), which belongs to w and ℝ3.

Secondly, for closure under vector addition, consider two points P1 = (a1t1, b1t1, c1t1) and P2 = (a2t2, b2t2, c2t2) in w. Their sum is (a1t1 + a2t2, b1t1 + b2t2, c1t1 + c2t2), which is another point on the line through the origin, therefore in w.

Finally, for closure under scalar multiplication, take any scalar k and point P = (at, bt, ct) in w. The product kP is (k*at, k*bt, k*ct), which is also on the line through the origin, so it remains in w.

Since w satisfies all three criteria, it is a subspace of ℝ3.

Answer 2

The subset w is a subspace of R^3 because it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication, satisfying the three necessary criteria for being a subspace.

Step-by-step explanation:

To show that w is a subspace of R^3, we must verify that it satisfies the three subspace criteria:

1. It contains the zero vector.
2. It is closed under vector addition.
3. It is closed under scalar multiplication.

The line is represented by parametric equations x = at, y = bt, and z = ct. For the first criterion, if we choose t = 0, we find that x = y = z = 0, which is the zero vector in R^3, and therefore w satisfies the first criterion.

For the second criterion, let's take two vectors on the line: v1 = (a·t1, b·t1, c·t1) and v2 = (a·t2, b·t2, c·t2). Under vector addition, v1 + v2 = (a·(t1+t2), b·(t1+t2), c·(t1+t2)), which is the same as (a, b, c) multiplied by the scalar (t1 + t2). This means that the sum is still on the line, thus w is closed under vector addition.

For the third criterion, if we take any scalar κ and multiply it with a vector v1, we get κ v1 = (κ a·t1, κ b·t1, κ c·t1), which can also be written as (a, b, c) multiplied by the new scalar κ·t1. Hence, w is closed under scalar multiplication.

Therefore, since w satisfies all three criteria, it is indeed a subspace of R^3.