Determine which of the following sets of vectors 𝐱=(x1, x2, …, xn)^T are subspaces of ℝ^n : (a)

step 1 So going through these, we can say that A is going to be a subspace. That's going to be the set

Determine which of the following sets of vectors 𝐱=(x1, x2,...

Determine which of the following sets of vectors $\mathbf{x}=\left(x_1, x_2, \ldots, x_n\right)^T$ are subspaces of $\mathbb{R}^n$ : (a) all equal entries $x_1=\cdots=x_n ;(b)$ all positive entries: $x_i \geq 0 ;(c)$ first and last entries equal to zero: $x_1=x_n=0 ;(d)$ entries add up to zero: $x_1+\cdots+x_n=0$; (e) first and last entries differ by one: $x_1-x_n=1$.

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Key Concepts

Definition of Subspace

A subspace of a vector space is a subset that is also a vector space under the same operations. This means it must contain the zero vector, and be closed under both vector addition and scalar multiplication.

Zero Vector Inclusion

One of the fundamental properties any subset must satisfy to be considered a subspace is that it contains the zero vector. If the zero vector is not present, then the subset cannot be a subspace.

Closure Under Addition

For a set to be a subspace, the sum of any two vectors in the set must also be a vector in the set. This property ensures that the set is stable under the addition operation defined in the vector space.

Closure Under Scalar Multiplication

A subspace must also be closed under scalar multiplication, meaning that multiplying any vector in the subset by any scalar from the field will yield another vector that is still within the subset.

Homogeneous vs. Inhomogeneous Conditions

Subsets defined by homogeneous linear equations (typically equal to zero) naturally pass through the origin and are candidates for subspaces, whereas subsets defined by inhomogeneous conditions (nonzero constants) do not contain the zero vector and therefore cannot be subspaces.

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