The Rank-Nullity Theorem for Linear Transformations and Matrices

Let $T:V\rightarrow W$ be a lin. trans. from a fin-dim. v.s. V to an arbitrary space W, then what does the Rank-Nullity Theorem state?

rank(T) + nullity(T) = dim(V)

Let V and W be vector spaces over a field K with dim(V) = dim(W) = n, then when is a lin. trans. $T:V\rightarrow W$ injective?

iff it is surjective

Let A be an m x n matrix in $M_{mn}\left(K\right)$, then what is the row space of A (Row(A))?

the subspace of $K^{n}$ spanned by the rows of A

Let A be an m x n matrix in $M_{mn}\left(K\right)$, then what is the column space of A (Col(A))?

the subspace of $K^{m}$ spanned by the columns of A

If $A,B\in M_{mn}\left(K\right)$ are row equivalent / column equivalent wht can we say about Row(B) and Col(B)?

Row(B) = Row(A)

Col(B) = Col(A)

For any matrix $A\in M_{mn}\left(K\right)$, what is the dimension of Row(A), and what is a basis for Row(A)?

-dim(Row(A)) is the number of non-zero rows in the RREF of A

-these non-zero rows form the basis for Row(A)

For any m x n matrix A, what is the null space of A (Null(A))?

$$Null(A)=\left\lbrace v\in K^{n}\vert Av=0\right\rbrace$$

What is dim(Null(A))? (=dim(Ker($T_{A}$)))

nullity(A) (=nullity($T_{A}$))

Let $A\in M_{mn}\left(K\right)$. Define $T_{A}:K^{n}\rightarrow K^{m}$ to be the lin. trans. given by $T_{A}\left(v\right)=Av$, then what does Col(A) equal?

$$Im\left(T_{A}\right)$$

What is dim(Col(A))? (=dim(Im($T_{A}$)))

rank(A) (=rank($T_{A}$))

If A is an m x n matrix, then what does the Rank-Nullity Theorem for matrices state?

rank(A) + nullity(A) = n

Let $A\in M_{mn}\left(K\right)$, then what does dim(Col(A)) equal and what does this imply about the rank($A^{T}$)?

-dim(Col(A)) = dim(Row(A))

-rank($A^{T}$) = rank(A)

Let $A\in M_{mn}\left(K\right)$, then when is $\lambda\in K$ an eigenvalue of A?

iff it is an eigenvalue of $A^{T}$