Linear Subspaces: Tests, Spans, Bases, and Operations

All flashcards

What is a linear subspace?

A subset of a vector space that is itself a vector space under the same addition and scalar multiplication.

State the two-vector subspace test.

A nonempty set $W$ is a subspace if $\alpha w_1+\beta w_2\in W$ for every $w_1,w_2\in W$ and every pair of scalars $\alpha,\beta$.

What are the trivial subspaces of $V$?

The zero subspace $\{0\}$ and the whole vector space $V$.

Why must every subspace contain zero?

A subspace is a vector space, so it contains its additive identity. Equivalently, the linear-combination test gives $0w_1+0w_2=0$.

Give an example of a coordinate subspace of $\mathbb R^3$.

The set $\{(x,y,z):z=0\}$ is the $xy$-plane through the origin and is closed under addition and scalar multiplication.

Quiz questions

1. How is the column space of a matrix $A$ defined?Answer: It is the span of the columns of $A$.
2. What is the span of a set of vectors?Answer: The smallest subspace containing all the vectors.
3. What two properties define a basis of a subspace?Answer: Its vectors are linearly independent and span the subspace.
4. For finite-dimensional subspaces $U$ and $W$, which formula gives the dimension of their sum?Answer: $\dim(U+W)=\dim U+\dim W-\dim(U\cap W)$
5. How can you test whether a vector $v$ lies in the span of given vectors?Answer: Solve the linear system asking whether their linear combination equals $v$.