Vector Subspaces, Linear Applications, and Quadratic Forms
A.1.7. Vector Subspace
Let S be a subset of a vector space E. S is a vector subspace if and only if:
- It contains the zero vector (0).
- It simultaneously meets these conditions:
- For any two vectors v1 and v2 belonging to S, v1 + v2 also belongs to S.
- For any vector v1 and any real number k, kv1 belongs to S.
The following are not vector subspaces:
- Numbers with exponents.
- Vectors defined by u = (x, y) where xy + x = 0 (product of two coordinates).
- Logarithms, etc. (certain mathematical constructs).
The following
Read MoreUnderstanding Megalithic Constructions: Menhirs and Dolmens
Megalithic Construction Types
1. Menhirs
The menhir is the simplest type of megalithic monument. It consists of a stone, of variable shape and height (1 to 6 m), fixed on the ground. Some menhirs stand isolated. The most significant example is in Locmariaquer (in Morbihan, France), which was a 23 m tall simple Menhir.
Menhirs may also appear clustered, distributed in rows, forming alignments, characteristic of Brittany. The best-known alignment is at Carnac, where thousands of standing stones are arranged
Read MoreLandform Evolution: Mountain Ranges, Erosion, and Coastal Features
R. jurásico
This landform can form in young mountain ranges. It consists of folds alternating between anticlines and synclines. Anticlines in the water create erosion valleys perpendicular to the summit (cluses) and parallel valleys (combos). Once erosion occurs, it empties fast, and former synclinal valleys are lifted (syncline hook). The erosion of the anticline valley exposes the anticline, and the cycle begins again.
R. apalachense
This landform forms covering a mountainous area that has been
Read MoreEarth’s Energy: Sources, Transfer, and Phenomena
Energy
Energy Transfer Mechanisms
- Convection: Heat transfer by mass movement or circulation within a substance.
- Radiation: Energy emitted by matter at a given temperature, produced directly from the source outwards in all directions.
- Conduction: The only way to transfer heat through solids.
Forms of Energy Entry and Exit on Earth
Reflection and Radiation
Winds
Winds are the movement of air in the atmosphere, especially in the troposphere, due to natural causes. It is a meteorological phenomenon. The cause
Read MoreUnderstanding Minerals and Rocks: Composition and Formation
Minerals: The Building Blocks of Rocks
Minerals are solid substances formed by the combination of chemical elements found in the Earth’s crust. Rocks are composed of minerals. To be classified as a mineral, a substance must possess the following characteristics:
- Be solid
- Be naturally occurring
- Have an inorganic origin
- Have a homogeneous chemical composition
Minerals are pure substances, each with a unique chemical composition. This composition determines many of its properties. However, impurities within
Read MoreOceanic Ridges: Formation, Characteristics, and Plate Tectonics
Oceanic Ridge Characteristics and Plate Tectonics
The oceanic ridge system is an undersea mountain range, rising between 2 and 3 km above the abyssal plain. It extends for approximately 65,000 km, periodically interrupted by laterally moving fractures called transform faults. The central depression is known as the rift valley.
Age and Sedimentation
The age of the ocean floor is less than 185 million years (Ma). No rocks currently exist within the ridges that are older than this. The older oceanic crust
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