Definition. Let V be a vector space and let H⊆V (H is a subset of V). We say that H is a subspace of V if all of the following are true:
- The zero vector 0 is in H.
- If u,v are in H, then u+v is in H. (Closure under addition)
- If c is any scalar and u is any vector in H, then cu is in H. (Closure under scalar multiplication)
H={0} (the set only containing the zero vector) is always a subset of any vector space V, and V is always a subset of itself.
Is R2 a subspace of R3? No, because elements of R2 are in the form (ab), and elements of R3 are in the form ⎝⎛cde⎠⎞, which cannot be equal to each other; therefore R2 is not even a subset of R3, let alone a subspace. However, you can find subspaces within R3 that resemble R2.
Ex. Verify that H={⎝⎛st0⎠⎞∣ s,t∈R} is a subspace of R3.
- Note H is a subset of R3.
- When s=t=0, we get the zero vector ⎝⎛000⎠⎞ in H.
- Let u=⎝⎛s1t10⎠⎞, v=⎝⎛s2t20⎠⎞ be in H. u+v=⎝⎛s1+s2t1+t20⎠⎞ so u+v is in H.
- Let c be any real number. cu=c⎝⎛s1t10⎠⎞=⎝⎛cs1ct10⎠⎞ so cu is in H.
Thus H is a subspace of R3.
Recall that C∞(R) is the set of infinitely differentiable functions on the reals.
Is P, the collection of all polynomials, a subspace of C∞(R)?
- Any polynomial can be differentiated as many times as you want, so P is a subset of C∞(R).
- f(x)=0 for all x is a polynomial, so 0 is in P.
- The sum of two polynomials is also a polynomial, so P is closed under addition.
- The product of a scalar and any polynomial is also a polynomial, so P is closed under scalar multiplication.
Thus P is a subspace of C∞(R).
Recall our example from before of H={⎝⎛st0⎠⎞∣ s,t∈R} being a subspace of R3.
H={s⎝⎛100⎠⎞+t⎝⎛010⎠⎞∣ s,t∈R}=span{e1,e2}.
The span of a set of vectors in V will always be a subspace of V.
Ex. Let H be the set of all vectors of the form ⎝⎜⎜⎛a−3bb−aab⎠⎟⎟⎞ Show that H is a subspace of R4.
⎝⎜⎜⎛a−3bb−aab⎠⎟⎟⎞=a⎝⎜⎜⎛1−110⎠⎟⎟⎞+b⎝⎜⎜⎛−3101⎠⎟⎟⎞H=span{⎝⎜⎜⎛1−110⎠⎟⎟⎞,⎝⎜⎜⎛−3101⎠⎟⎟⎞}
Since the span of a collection of vectors from R4 is a subspace of R4, then H is a subspace of R4.