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The set is considered a subspace because vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication. This set is the span of (1,0,−1), a line through the origin.

Conditions: The vectors are of the form (x,0,-x).; The operation is vector addition or scalar multiplication.

The set is considered a subspace because vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication. This set is the span of (1,0,−1), a line through the origin.

Conditions: The vectors are of the form (x,0,-x).; The operation is vector addition or scalar multiplication.

Scalar multiplication preserves the relation because multiplying the equation $x_3 = x_1 + x_2$ by a scalar $c$ yields $cx_3 = cx_1 + cx_2$. This shows that the third component of the scaled vector $cx$ equals the sum of its first and second components.

Conditions: $x$ is a vector in the set satisfying $x_3 = x_1 + x_2$.; $c$ is a real scalar.; Scalar multiplication is defined componentwise.

Scalar multiplication preserves the relation because multiplying the equation $x_3 = x_1 + x_2$ by a scalar $c$ yields $cx_3 = cx_1 + cx_2$. This shows that the third component of the scaled vector $cx$ equals the sum of its first and second components.

Conditions: $x$ is a vector in the set satisfying $x_3 = x_1 + x_2$.; $c$ is a real scalar.; Scalar multiplication is defined componentwise.

Adding two vectors preserves the relation because the sum of the third components equals the sum of the first components plus the sum of the second components. Specifically, if $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$, then $(x+y)_3 = x_3 + y_3 = (x_1 + x_2) + (y_1 + y_2) = (x_1 + y_1) + (x_2 + y_2) = (x+y)_1 + (x+y)_2$.

Conditions: $x$ and $y$ are vectors in $\mathbb{R}^3$.; $x$ and $y$ satisfy the equation $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$ respectively.; Vector addition is defined componentwise.

Adding two vectors preserves the relation because the sum of the third components equals the sum of the first components plus the sum of the second components. Specifically, if $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$, then $(x+y)_3 = x_3 + y_3 = (x_1 + x_2) + (y_1 + y_2) = (x_1 + y_1) + (x_2 + y_2) = (x+y)_1 + (x+y)_2$.

Conditions: $x$ and $y$ are vectors in $\mathbb{R}^3$.; $x$ and $y$ satisfy the equation $x_3 = x_1 + x_2$ and $y_3 = y_1 + y_2$ respectively.; Vector addition is defined componentwise.