Final answer:
The set of all vectors of the form [1,x,y]t, where x and y range over all real numbers, is a subspace of R3 because it satisfies the three conditions for a subspace: containing the zero vector, closure under addition, and closure under scalar multiplication.
Step-by-step explanation:
In order to determine if the set of all vectors of the form [1,x,y]t, where x and y range over all real numbers, is a subspace of R3, we need to check if it satisfies the three conditions for a subspace:
- The set must contain the zero vector.
- The set must be closed under addition.
- The set must be closed under scalar multiplication.
First, let's check if the zero vector [0,0,0] is in the set. Since the zero vector is of the form [1, 0, 0]t, we can see that it is indeed in the set.
Next, let's check closure under addition. Take two vectors from the set, [1, x1, y1]t and [1, x2, y2]t. Their sum is [2, x1 + x2, y1 + y2]t. Since x1, x2, y1, and y2 are all real numbers, the sum is also of the form [1, x, y]t and thus in the set.
Finally, let's check closure under scalar multiplication. Take a vector from the set, [1, x, y]t, and multiply it by a scalar c. The result is [c, cx, cy]t, which is still in the set since cx and cy are real numbers.
Therefore, the set of all vectors of the form [1,x,y]t, where x and y range over all real numbers, is indeed a subspace of R3.