Anki Deck Changes

Note 1: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: (scS~v1D#

modified

Before

Front

\((AB)^{\top}=B^\top A^\top\)

Back

\((AB)^{\top}=B^\top A^\top\)

After

Front

\((AB)^{\top}=\)\(B^\top A^\top\)

Back

\((AB)^{\top}=\)\(B^\top A^\top\)

Field-by-field Comparison

Field	Before	After
Text	\((AB)^{\top}={{c1::B^\top A^\top}}\)	\((AB)^{\top}=\){{c1::\(B^\top A^\top\)}}

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::3._Row_space_and_transpose

Note 2: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: B:N,l}NO-6

added

New Note

Front

The rank of \(A\) is \(0\) if and only if \(A\) is the zero matrix.

Back

The rank of \(A\) is \(0\) if and only if \(A\) is the zero matrix.

Field-by-field Comparison

Field	Before	After
Text		The rank of \(A\) is {{c2::\(0\)}} if and only if {{c1::\(A\) is the zero matrix}}.

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::2._Column-space_and_rank

Note 3: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: CXkeb-2S`g

added

New Note

Front

Let \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) be a linear transformation. There is a?

Back

Let \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) be a linear transformation. There is a?

There is a unique \(m \times n\) matrix A such that \(T = T_A\) meaning that \(T(x) = T_A(x) = Ax\) for all \(x \in \mathbb{R}^n\).

Field-by-field Comparison

Field	Before	After
Front		Let \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) be a linear transformation. There is a?
Back		There is a unique \(m \times n\) matrix A such that \(T = T_A\) meaning that \(T(x) = T_A(x) = Ax\) for all \(x \in \mathbb{R}^n\).

Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::3._The_matrix_of_a_linear_transformation

Note 4: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: D&_[T~I?f`

added

New Note

Front

Which vector is always in the nullspace of \(A\)?

Back

Which vector is always in the nullspace of \(A\)?

The zero vector \(0\)

Field-by-field Comparison

Field	Before	After
Front		Which vector is always in the nullspace of \(A\)?
Back		The zero vector \(0\)

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace

Note 5: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: Dd-0>Kd049

added

New Note

Front

The columns of \(A\) are independent if and only if \(x = 0\) is the only vector for which \(Ax = 0\) (Linear combination view).

Back

The columns of \(A\) are independent if and only if \(x = 0\) is the only vector for which \(Ax = 0\) (Linear combination view).

Field-by-field Comparison

Field	Before	After
Text		The columns of \(A\) are independent if and only if {{c1::\(x = 0\) is the  only vector for which \(Ax = 0\)}} (Linear combination view).

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication

Note 6: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: D|B,=vwf}e

added

New Note

Front

For matrices A, B, C, \(A(B+C)=\)\(AB + BC\) (distributivity)

Back

For matrices A, B, C, \(A(B+C)=\)\(AB + BC\) (distributivity)

Field-by-field Comparison

Field	Before	After
Text		For matrices A, B, C, \(A(B+C)=\){{c1::\(AB + BC\) (distributivity)}}

Tags: ETH::1._Semester::LinAlg::2._Matrices::3._Matrix_multiplication::2._Definition_and_basic_properties

Note 7: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: J<060JA%BR

added

New Note

Front

Can a nilpotent matrix have an inverse?

Back

Can a nilpotent matrix have an inverse?

No, as the \(0\) matrix does not have an inverse.

Field-by-field Comparison

Field	Before	After
Front		Can a nilpotent matrix have an inverse?
Back		No, as the \(0\) matrix does not have an inverse.

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations

Note 8: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: JlD}ITLxEy

added

New Note

Front

What is the composition of two linear transformations \(T_A \circ T_B\)?

Back

What is the composition of two linear transformations \(T_A \circ T_B\)?

\(T_A \circ T_B = T_{AB}\)

Field-by-field Comparison

Field	Before	After
Front		What is the composition of two linear transformations \(T_A \circ T_B\)?
Back		\(T_A \circ T_B = T_{AB}\)

Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::3._The_matrix_of_a_linear_transformation

Note 9: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: K4@L>.#ir<

added

New Note

Front

For all matrices \(A\), \((A^\top)^\top = \)\(A\).

Back

For all matrices \(A\), \((A^\top)^\top = \)\(A\).

Field-by-field Comparison

Field	Before	After
Text		For all matrices \(A\), \((A^\top)^\top = \){{c1::\(A\)}}.

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::3._Row_space_and_transpose

Note 10: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: KZs-,vID&:

added

New Note

Front

What are kernel and image of a linear transformation?

Back

What are kernel and image of a linear transformation?

The kernel is the nullspace and the image the column space.

Field-by-field Comparison

Field	Before	After
Front		What are kernel and image of a linear transformation?
Back		The <b>kernel</b> is the <b>nullspace</b> and the <b>image</b> the <b>column space</b>.

Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::4._Kernel_and_Image

Note 11: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: QM2twywAT5

added

New Note

Front

Do the rank or independent columns change if we re-order the columns?

Back

Do the rank or independent columns change if we re-order the columns?

The independent columns change, but not their number and thus not the rank.

Field-by-field Comparison

Field	Before	After
Front		Do the rank or independent columns change if we re-order the columns?
Back		The independent columns change, but not their number and thus not the rank.

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::2._Column-space_and_rank

Note 12: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: eUCQYkiYf@

modified

Before

Front

What is a property that always hold for linear transformations?

Back

What is a property that always hold for linear transformations?

for a linear transformation \(T(X)\) \(T(0) =0\)

After

Front

What is a property that always hold for linear transformations?

Back

What is a property that always hold for linear transformations?

for a linear transformation \(T(X)\): \(T(0) = 0\)

Field-by-field Comparison

Field	Before	After
Back	for a linear transformation \(T(X)\) \(T(0) =0\)	for a linear transformation \(T(X)\): \(T(0) = 0\)

Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals

Note 13: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: l_;^0p%n!C

added

New Note

Front

Every matrix transformation is a linear transformation.

Back

Every matrix transformation is a linear transformation.

The inverse is also true.

Field-by-field Comparison

Field	Before	After
Text		Every matrix transformation is a {{c1:: linear transformation}}.
Extra		The inverse is also true.

Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::2._Linear_transformations_and_linear_functionals

Note 14: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: lgnIL]HR}%

added

New Note

Front

What can we use to speed up long matrix multiplications, for example \(w^\intercal (vw^\intercal) v\)?

Back

What can we use to speed up long matrix multiplications, for example \(w^\intercal (vw^\intercal) v\)?

We can use associativity: \(w^\intercal (vw^\intercal) v = (w^\intercal v)(w^\intercal v)\).

Field-by-field Comparison

Field	Before	After
Front		What can we use to speed up long matrix multiplications, for example \(w^\intercal (vw^\intercal) v\)?
Back		We can use associativity: \(w^\intercal (vw^\intercal) v = (w^\intercal v)(w^\intercal v)\).

Tags: ETH::1._Semester::LinAlg::2._Matrices::3._Matrix_multiplication::4._Everything_is_matrix_multiplication

Note 15: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: nCym|+n@l+

added

New Note

Front

We can view the matrix-vector product Ax in two ways:

Back

We can view the matrix-vector product Ax in two ways:

row view: result is a vector where each entry is the scalar product of row \(i\) of \(A\) with \(x\): \((Ax)_{i} = A_i^\top x\)
column view: The resulting vector is a linear combination of the columns of \(A\)

Field-by-field Comparison

Field	Before	After
Front		We can view the matrix-vector product Ax in two ways:
Back		<ul><li>row view: result is a vector where each entry is the scalar product of row \(i\) of \(A\) with \(x\): \((Ax)_{i} = A_i^\top x\)</li><li>column view: The resulting vector is a linear combination of the columns of \(A\)</li></ul>

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication

Note 16: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: p,i>w2r!)Y

added

New Note

Front

A matrix \(A\) is nilpotent if {{c1:: there is a \(k \in \mathbb{N}\) such that \(A^k = 0\)}}.

Back

A matrix \(A\) is nilpotent if {{c1:: there is a \(k \in \mathbb{N}\) such that \(A^k = 0\)}}.

\(A\) cannot have an inverse.

Field-by-field Comparison

Field	Before	After
Text		A matrix \(A\) is {{c2::nilpotent}} if {{c1:: there is a \(k \in \mathbb{N}\) such that \(A^k = 0\)}}.
Extra		\(A\) cannot have an inverse.

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations

Note 17: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: sQOMX5~Sf=

added

New Note

Front

Matrix multiplication is not commutative most of the time.

Back

Matrix multiplication is not commutative most of the time.

Field-by-field Comparison

Field	Before	After
Text		Matrix multiplication is {{c1::not}} commutative most of the time.

Tags: ETH::1._Semester::LinAlg::2._Matrices::3._Matrix_multiplication::2._Definition_and_basic_properties

Note 18: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: w49b;wY}uY

added

New Note

Front

The {{c2::rank of a matrix \(\textbf{rank}(A), A \in \mathbb{R}^{m \times n}\) is a number between 0 and n}} which .

Back

The {{c2::rank of a matrix \(\textbf{rank}(A), A \in \mathbb{R}^{m \times n}\) is a number between 0 and n}} which .

A column is independent if it is not the linear combination of the previous ones (or the next ones, if you do it the other way round).

Field-by-field Comparison

Field	Before	After
Text		The {{c2::rank of a matrix \(\textbf{rank}(A), A \in \mathbb{R}^{m \times n}\) is a number between 0 and n}} which {{c1:counts the number of independent columns}}.
Extra		<div>A column is independent if it is not the linear combination of the <b>previous ones</b> (or the <b>next ones</b>, if you do it the other way round).</div>

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::1._Matrix-Vector_multiplication

Note 19: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: z{mV]q!JSS

added

New Note

Front

How do we find the matrix \(A\) associated with a linear transformation \(T_A\)?

Back

How do we find the matrix \(A\) associated with a linear transformation \(T_A\)?

For \(e_1, e_2, \dots, e_n\) we calculate \(T_A(e_k)\) to find the \(k\)-th column of \(A\): \[ A = \begin{bmatrix} | & | & \text{} & | \\ T(e_1) & T(e_2) & \dots & T(e_n) \\ | & | & \text{ } & | \end{bmatrix} \]

Field-by-field Comparison

Field	Before	After
Front		How do we find the matrix \(A\) associated with a linear transformation \(T_A\)?
Back		For \(e_1, e_2, \dots, e_n\) we calculate \(T_A(e_k)\) to find the \(k\)-th column of \(A\): \[ A = \begin{bmatrix} \| & \| & \text{} & \| \\ T(e_1) & T(e_2) & \dots & T(e_n) \\ \| & \| & \text{ } & \| \end{bmatrix} \]<br>

Tags: ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::3._The_matrix_of_a_linear_transformation

Note 20: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: z~{N,b0:-A

added

New Note

Front

The independent columns of \(A\) {{c1::span the columns space \(\textbf{C}(A)\) of \(A\)}}. Lemma 2.11

Back

The independent columns of \(A\) {{c1::span the columns space \(\textbf{C}(A)\) of \(A\)}}. Lemma 2.11

Proven by induction, adding elements that are a linear combination of other ones doesn't change span, thus we can iteratively remove the dependent columns.

Field-by-field Comparison

Field	Before	After
Text		The {{c2::independent}} columns of \(A\) {{c1::span the columns space \(\textbf{C}(A)\) of \(A\)}}. Lemma 2.11
Extra		Proven by induction, adding elements that are a linear combination of other ones doesn't change span, thus we can iteratively remove the dependent columns.

Tags: ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::2._Column-space_and_rank

Note 1: ETH::LinAlg

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Note 11: ETH::LinAlg

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Note 12: ETH::LinAlg

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Note 13: ETH::LinAlg

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Note 14: ETH::LinAlg

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Note 16: ETH::LinAlg