Math 210

Basis for degree-n polynomials

Idea: The space Pn​ = all polynomials a0​+a1​x+…+an​x^n.
Basis: B={1,x,x^2,…,x^n} because these are independent and span all degree-≤ n polynomials.
Dimension: n+1; coordinates = column vector (a0​,a1​,…,an​)T.

Let V = span( w ) where w is a nonzero vector in Rn. Is V a subspace of Rn?. What are all the subspaces

of V ?

• V=span(w) (with w≠0) is a 1-dimensional subspace (a line through the origin) because it’s closed under addition and scalar multiplication.

• The only subspaces of V are {0⃗} and V itself.

Rank ↔ Free Variables

(a) In RREF, each pivot fixes one variable; non-pivot columns are free.
So if rank = r → n–r free variables.
(b) Conversely, if there are n–r free variables, then exactly r pivots → rank = r.

Consider a linear system…

The coefficients correspond to the m×n matrix

(a) What must the rank of A be if, for whichever d1,…,dm​ you choose, the linear system always has a solution?
(b) If m≠n, is it possible for the system to have a unique solution for any choice of d1,…,dm?

(a) For a solution to exist for every right-hand side ddd:
→ The columns of A must span all of Rm.
→ Therefore, rank(A) = m.

(b) For a unique solution for every d:
→ We need rank(A) = n, meaning the columns are linearly independent.
→ This can only happen if m=n (square, invertible system).
→ If m≠n, it’s not possible to have a unique solution for all d.

8. Let A be an m × n matrix and B be the matrix we get when we convert A into row echelon form.

• Is the span of the column vectors of A the same as the span of the column vectors of B?

• Is the span of the row vectors of A the same as the span of the row vectors of B?

Column vectors: ❌ Not necessarily the same.

Row operations can change the column relationships — they can alter the column space.

Row vectors: ✅ Always the same.

Row operations do not change the span of the rows, so Row(A) = Row(B).

Linearity check (add + scale)

T is linear iff T(u+v)=T(u)+T(v) and T(cv)=cT(v); equivalently T(0)=0.

• T(p(x))=xp(x): linear (both rules hold).

• T(A)=A⊤: linear (transpose preserves add & scalar).

V → W be a linear transformation with T (⃗v 1). Independence under T

If the images wi​ are independent, then the vi must also be independent. If the vi are independent, the wi may or may not be (independence can be lost).

V → W be a linear transformation with T (v 1)Spanning under T

If vi span V, the wi=T(vi) span the range of T . If wi span W, the vi​ might not span V.

Let T : V → W be a linear transformation that is NOT one-to-one.
Show that the nullspace of T contains vectors other than the zero vector 0.

If T is not one-to-one, then different inputs map to the same output.

That means there exists some v ≠ 0 such that T(v) = 0.

→ So the nullspace (set of all v where T(v) = 0) contains nonzero vectors.

Nullity = # of free variables

In RREF, pivot columns = rank; non-pivot = free variables.

Each free variable adds one dimension to the nullspace → nullity = # free variables.