# Notes - Linear Algebra MT22, Misc

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/notes/misc/ · Updated: 2024-07-07 · Tags: uni, notes

- [Course - Linear Algebra MT22](https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/)

### 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.

@Define an isomorphism.::
An invertible map.

What’s a quick proof that every matrix whose 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
$$
EA = I_n \iff E = A^{-1}
$$

@State the four conditions for a matrix to be in RREF.::
1. Every zero row is at the bottom of the matrix.
2. Every leading entry in a row is a $1$.
3. Any column containing a $1$ has all other entries equal to $0$.
4. 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.::
$$
\forall a, b \in X, f(a) = f(b) \implies a = b
$$

Let $f : X \to Y$. @Define what it means for $f$ to be surjective.::
$$
\forall y \in Y, \text{ } \exists x \in X \text{ s.t. } f(x) = y
$$

If $T$ is a projection (i.e. $T^2 = T$), then what is true about $I - T$?::
It is also a projection.

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