Notes - Linear Algebra MT22, Misc
Flashcards
How do you show that an infinite set of vectors is linearly independent?
Show that every finite subset is linearly independent.
When considering linear combinations and infinite sets, what is always true?
Linear combinations only have finitely many terms.
Why isn’t the vector space of all real sequences $(a, b, c,\ldots)$ spanned by the set $\{(1,0,0,\ldots), (0,1,0,\ldots),\ldots\}$?
Because a vector like $(1, 1, 1,\ldots)$ would need to be the linear combination of all basis vectors, but linear combinations always involve a finite number of terms.
Given a vector space of dimension $n$, what is true about any linearly independent set of size $n$?
It forms a basis for that vector space.
What is an isomorphism?
An invertible map.
What’s a quick proof that every matrix who’s RREF is the identity matrix has an inverse?
Let $E = E _ n \ldots E _ 1$, where each $E _ i$ is an elementary row operation. Then
\[EA = I_n \iff E = A^{-1}\]What are the four conditions of a matrix being in RREF?
- Every zero row is at the bottom of the matrix.
- Every leading entry in a row is a $1$.
- Any column containing a $1$ has all other entries equal to $0$.
- All leading $1$s are to the right of all leading $1$s above them.
Let $f : X \to Y$. Define what it means for $f$ to be injective.
Let $f : X \to Y$. Define what it means for $f$ to be surjective.
If $T$ is a projection (i.e. $T^2 = T$), then what is true about $I - T$?
It is also a projection.