6 Linear Transformations Free Online FlashCards

What condition defines a linear transformation?

A linear transformation preserves linear combinations: for scalars $a,b$, $T(a\mathbf{u}+b\mathbf{v})=aT(\mathbf{u})+bT(\mathbf{v})$.

What must a linear transformation do to the zero vector?

Every linear transformation maps the zero vector to the zero vector: $T(\mathbf{0})=\mathbf{0}$.

Why does a basis determine a linear transformation?

A linear transformation is completely determined by its values on a basis, because every vector is a linear combination of the basis vectors.

What is the kernel of a linear transformation?

The kernel is $\ker T=\{\mathbf{v}:T(\mathbf{v})=\mathbf{0}\}$, the set of inputs sent to the zero vector.

What is the image of a linear transformation?

The image is $\operatorname{im}T=\{T(\mathbf{v}):\mathbf{v}\in V\}$, the set of all attainable outputs.

How is a matrix transformation’s kernel found?

For a matrix transformation $T_A(\mathbf{x})=A\mathbf{x}$, the kernel is the null space $\{\mathbf{x}:A\mathbf{x}=\mathbf{0}\}$.

What is the image of a matrix transformation?

For $T_A(\mathbf{x})=A\mathbf{x}$, the image is the column space of $A$, spanned by the columns of $A$.

When is a linear transformation injective?

A linear transformation is injective exactly when its kernel is trivial: $\ker T=\{\mathbf{0}\}$.

When is a linear transformation surjective?

A linear transformation is surjective exactly when its image equals the entire codomain: $\operatorname{im}T=W$.

What does each column of a relative matrix representation contain?

The matrix $[T]_{\mathcal{C}\leftarrow\mathcal{B}}$ has column $j$ equal to $[T(\mathbf{v}_j)]_{\mathcal{C}}$.

What matrix represents a composition of transformations?

If $A$ represents $U$ and $B$ represents $T$, then $[T\circ U]=BA$: apply $A$ first, then $B$.

What does a change-of-coordinate matrix do?

If $P=P_{\mathcal{B}\leftarrow\mathcal{B}'}$, then $[\mathbf{v}]_{\mathcal{B}}=P[\mathbf{v}]_{\mathcal{B}'}$. Its columns contain the new basis vectors in old coordinates.