$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
FullPage
result
Concepts
If
Only If
proof

Injective and Surjective Conditions

Let $T:V\to W$, where $V$ and $W$ are both vector spaces and $V$ is finite dimensional. Then $T$ is injective if and only if the nullity of $T$ is $0$, and $T$ is surjective if and only if the rank of $T$ is the dimensional of $W$.

Concepts

If the dimension of $W$ is infinite, then the rank of $T$ equalling the rank of $W$ does not necessarly imply that $T$ is subjective.

If

Coming soon

Only If

Coming soon

Proof

Coming soon

Injective and Surjective Conditions

Let $T:V\to W$, where $V$ and $W$ are both vector spaces and $V$ is finite dimensional. Then $T$ is injective if and only if the nullity of $T$ is $0$, and $T$ is surjective if and only if the rank of $T$ is the dimensional of $W$.

Concepts

If the dimension of $W$ is infinite, then the rank of $T$ equalling the rank of $W$ does not necessarly imply that $T$ is subjective.

If

Coming soon

Only If

Coming soon

Proof

Coming soon
FullPage
result
concepts
If
Only If
proof
FullPage
result
Concepts
If
Only If
proof

Injective and Surjective Conditions

Let $T:V\to W$, where $V$ and $W$ are both vector spaces and $V$ is finite dimensional. Then $T$ is injective if and only if the nullity of $T$ is $0$, and $T$ is surjective if and only if the rank of $T$ is the dimensional of $W$.

Concepts

If the dimension of $W$ is infinite, then the rank of $T$ equalling the rank of $W$ does not necessarly imply that $T$ is subjective.

If

Coming soon

Only If

Coming soon

Proof

Coming soon

Injective and Surjective Conditions

Let $T:V\to W$, where $V$ and $W$ are both vector spaces and $V$ is finite dimensional. Then $T$ is injective if and only if the nullity of $T$ is $0$, and $T$ is surjective if and only if the rank of $T$ is the dimensional of $W$.

Concepts

If the dimension of $W$ is infinite, then the rank of $T$ equalling the rank of $W$ does not necessarly imply that $T$ is subjective.

If

Coming soon

Only If

Coming soon

Proof

Coming soon
FullPage
result
concepts
If
Only If
proof