Let H be a subset of a vector space V. We have two main methods of determining if H is a subspace of V:
- Method I:
- Determine that 0 is in H
- Determine that if u,v are in H, then u+v is also in H (closed under addition)
- Determine that if u is in H and c is any scalar, then cu is also in H (closed under scalar multiplication)
- Method II:
- Find a set of vectors in V that span H.
- "Theorem. The span of any collection of vectors from V is a subspace of V."
Null and Column Spaces
There are some “natural” subspaces that are worth examining:
Let A be an m×n matrix. Recall that x→Ax is a linear transformation from Rn to Rm.
Claim NulA={x:Ax=0∣x in Rn} (the null space of A) is a subspace of Rn:
- A0=0, so 0 is in NulA.
- If u,v are in NulA, then u+v is also in NulA.
- Au=0
- Av=0
- A(u+v)=Au+Av=0+0=0
- If u is in NulA and c is any scalar, then cu is also in NulA.
- Au=0
- A(cu)=c(Au)=c(0)=0
Thus NulA is a subspace of Rn.
x→Ax is one-to-one iff Ax=0 has only the trivial solution. This is equivalent to NulA containing only the zero vector.
Column Space
Claim ColA=span{a1,...,an} where A=[a1,...,an] (the column space of A) is a subspace of Rm:
- a1,...,an are all vectors in Rm, thus ColA=span{a1,...,an} is a subspace of Rm by the previously stated theorem.
If v is in Rm, then how would we determine if there is some x in Rm such that Ax=v?
We’d determine if v is in the span of the columns of A, which is equivalent to v being in ColA. Thus ColA can be thought of as the image of Rm under the mapping x→Ax.
Ex. Find a spanning set for NulA where A=⎝⎛−3126−2−4−125138−7−1−4⎠⎞ This is equivalent to solving Ax=0.
⎝⎜⎜⎜⎜⎛x1x2x3x4x5⎠⎟⎟⎟⎟⎞→⎝⎛−3126−2−4−125138−7−1−4000⎠⎞→⎝⎛1−32−26−42−15318−1−7−4000⎠⎞→⎝⎛100−2002513102−1−10−2000⎠⎞→⎝⎛100−200210320−1−20000⎠⎞→⎝⎛100−200010−1203−20000⎠⎞=⎝⎜⎜⎜⎜⎛2x2+x4−3x5x2−2x4+2x5x4x5⎠⎟⎟⎟⎟⎞=x2⎝⎜⎜⎜⎜⎛21000⎠⎟⎟⎟⎟⎞+x4⎝⎜⎜⎜⎜⎛10−210⎠⎟⎟⎟⎟⎞+x5⎝⎜⎜⎜⎜⎛−30201⎠⎟⎟⎟⎟⎞
Thus the null space is spanned by these three vectors.
Ex. Let T:R3→R3 be T⎝⎛xyz⎠⎞=⎝⎛xy0⎠⎞
The null space of the standard matrix of T is {⎝⎛00z⎠⎞∣z in R}
The column space of the standard matrix of T is the entire xy plane.