Anki Deck Changes

Commit: 8c0bb4f1 - fixing my own retardation

Author: lhorva <lhorva@student.ethz.ch>

Date: 2026-01-02T14:29:50+01:00

Changes: 1 note(s) changed (0 added, 1 modified, 0 deleted)

Note 1: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: gU%jisb2z3
modified

Before

Front

ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties

Let \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar:
  1. \(v \cdot w = w \cdot v\) (symmetry / commutativity)
  2.  \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
  3.  \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
  4.  \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definedness)

Back

ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties

Let \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar:
  1. \(v \cdot w = w \cdot v\) (symmetry / commutativity)
  2.  \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
  3.  \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
  4.  \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definedness)

After

Front

ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties

Let \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar:
  1. \(v \cdot w = w \cdot v\) (symmetry / commutativity)
  2.  \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
  3.  \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
  4.  \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness)

Back

ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
Scalar product properties

Let \(u, v, w \in \mathbb{R}^m\) be vectors and \(\lambda \in \mathbb{R}\) a scalar:
  1. \(v \cdot w = w \cdot v\) (symmetry / commutativity)
  2.  \((\lambda v) \cdot w = \lambda (v \cdot w)\) (scalars move freely)
  3.  \(u \cdot (v + w) = u \cdot v + u \cdot w\) and \((u + v) \cdot w = u\cdot w + v \cdot w\) (distributivity)
  4.  \(v \cdot v \geq 0\) with equality if and only if \(v = 0\) (positive definiteness)
Field-by-field Comparison
Field Before After
Text Scalar product properties<br><br>Let&nbsp;\(u, v, w \in \mathbb{R}^m\)&nbsp;be vectors and&nbsp;\(\lambda \in \mathbb{R}\)&nbsp;a scalar:<br><ol><li>{{c1::\(v \cdot w = w \cdot v\)&nbsp;(symmetry / commutativity)}}</li><li>{{c2::&nbsp;\((\lambda v) \cdot w = \lambda (v \cdot w)\)&nbsp;(scalars move freely)}}</li><li>{{c3::&nbsp;\(u \cdot (v + w) = u \cdot v + u \cdot w\)&nbsp;and&nbsp;\((u + v) \cdot w = u\cdot w + v \cdot w\)&nbsp;(distributivity)}}</li><li>{{c4::&nbsp;\(v \cdot v \geq 0\)&nbsp;with equality if and only if&nbsp;\(v = 0\)&nbsp;(positive definedness)}}</li></ol> Scalar product properties<br><br>Let&nbsp;\(u, v, w \in \mathbb{R}^m\)&nbsp;be vectors and&nbsp;\(\lambda \in \mathbb{R}\)&nbsp;a scalar:<br><ol><li>{{c1::\(v \cdot w = w \cdot v\)&nbsp;(symmetry / commutativity)}}</li><li>{{c2::&nbsp;\((\lambda v) \cdot w = \lambda (v \cdot w)\)&nbsp;(scalars move freely)}}</li><li>{{c3::&nbsp;\(u \cdot (v + w) = u \cdot v + u \cdot w\)&nbsp;and&nbsp;\((u + v) \cdot w = u\cdot w + v \cdot w\)&nbsp;(distributivity)}}</li><li>{{c4::&nbsp;\(v \cdot v \geq 0\)&nbsp;with equality if and only if&nbsp;\(v = 0\)&nbsp;(positive definiteness)}}</li></ol>
Tags: ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::2._Scalar_multiplication
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