Linear Transformations

What is a linear transformation from a v.s. V to a v.s W over K?

a function T:V\rightarrow W s.t. \forall u,v\in V,\lambda\in K, we have

-T\left(u+v\right)=T\left(u\right)+T\left(v\right)

-T\left(\lambda u\right)=\lambda T\left(u\right)

What is the kernel of a linear transformation?

Ker\left(T\right)=\left\lbrace v\in V:T\left(v\right)=0\right\rbrace

If T:V\rightarrow W is a linear transformation, what can we say about Ker(T) and Im(T)?

-Ker(T) is a subspace of V

-Im(T) is a subspace of W

What is the nullity of T?

dim(Ker(T))

What is the rank of T?

dim(Im(T))

If T:V\rightarrow Wis a linear transformation, then when is T injective?

iff Ker\left(T\right)=\left\lbrace0\right\rbrace

When is a linear transformation T:U\rightarrow V called an isomorphism?

if it is a bijection

If we have any finite-dimensional v.s. V over a field K with dim(V) = n, what is it isomorphic to?

K^{n}

Let U and V be fin. dim. v.s. over K with basis B and C respectively, and let T:U\rightarrow V be a lin. trans., then what does _{B}\left\lbrack T\right\rbrack_{C}\left\lbrack u\right\rbrack_{B} equal?

\left\lbrack T\left(u\right)\right\rbrack_{C}

Let U, V, and W be fin-dim. v.s. over a field K with bases B, C and D respectively and let T:U\rightarrow V and S:V\rightarrow W be linear transformations, then what does _{B}\left\lbrack SoT\right\rbrack_{D} equal?

_{C}\left\lbrack S\right\rbrack_{D}\left._{B}\left\lbrack\right.T\right\rbrack_{C}

Let V and W be fin-dim. v.s. of the same dim. n over a field K, with bases B and C respectively. Let T:V\rightarrow W be a linear transformation, then when is T an isomorphism?

iff _{B}\left\lbrack T\right\rbrack_{C} is an invertible matrix (_{C}\left\lbrack T^{-1}\right\rbrack_{B}=\left(_{B}\left\lbrack T\right\rbrack_{C}\right)^{-1} )

Let V b e a fin-dim. v.s. with bases B and C. Let T:V\rightarrow V be a lin. trans., then what is_{}\left\lbrack T\right\rbrack_{B} equal to?

P_{B\rightarrow C}^{-1}\left\lbrack T\right\rbrack_{C}P_{B\rightarrow C}

If A,B\in M_{n}\left(K\right), when do we say that A and B are similar?

if \exists an invertible matrix P s.t. A=P^{-1}BP