Define subspace of a vector space. Let \(H = \left \{ \begin{pmatrix}s\\ t\\ 0\end{pmatrix}:s,t \in R \right \}\). Show that H is a subspace of:

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Let V be a vector space over a field k, then a non empty subset w of vector space v is saied to be subspace of v if it statisfies the following properties

  1. ∀ u, v ∈ W,   u + v ∈ w  [Closure under addition]
  2. ∀ u ∈ W,   c ∈ w, cu ∈ w  [Closure under scalar multiplication]
  3. The zero vector is in w i.e. 0 ∈ w


H = \(\left\{ \begin{bmatrix}s\\ t\\ 0\end{bmatrix} \right.\)     s, t ∈ R

To show H is a subspace of R3. First we check, if H is a subsset of R3.

Obviously, H ⊆ R3


* Closure Under Addition

Let H1 = \(\begin{bmatrix}s1\\ t1\\ 0\end{bmatrix}\), H2 = \(\begin{bmatrix}s2\\ t2\\ 0\end{bmatrix}\)  ∈ H

∴ H1 + H2 = \(\begin{bmatrix}s1 + s2\\ t1 + t2\\ 0\end{bmatrix}\) also ∈ H

So, it is closed under addition

* Closure under scalar multiplication

Here, H = \(\begin{bmatrix}s\\ t\\ 0\end{bmatrix}\)

Let, c ∈ H

∴ cH = c \(\begin{bmatrix}s\\ t\\ 0\end{bmatrix}\) = \(\begin{bmatrix}cs\\ ct\\ 0\end{bmatrix}\) ∈ H

so, it closed under scalar multiplication

* Obviously

0 = \(\begin{bmatrix}0\\ 0\\ 0\end{bmatrix}\) ∈ H

Hence, H is a subset of R3

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