Linear Algebra - Revision history

Wordpad: BloomWiki: Linear Algebra

2026-04-25T01:53:18Z

BloomWiki: Linear Algebra

← Older revision		Revision as of 01:53, 25 April 2026
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			<div style="background-color: #4B0082; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
	{{BloomIntro}}		{{BloomIntro}}
	Linear Algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. If algebra is the study of single numbers, linear algebra is the study of '''Grids''' and '''Arrows'''. It is the "Engine Room" of modern technology. Every time you filter a photo on Instagram, search Google, or use an AI like ChatGPT, millions of linear algebra calculations are happening in the background. It allows us to solve thousands of equations simultaneously, making it the most practical and widely used math in the 21st century.		Linear Algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. If algebra is the study of single numbers, linear algebra is the study of '''Grids''' and '''Arrows'''. It is the "Engine Room" of modern technology. Every time you filter a photo on Instagram, search Google, or use an AI like ChatGPT, millions of linear algebra calculations are happening in the background. It allows us to solve thousands of equations simultaneously, making it the most practical and widely used math in the 21st century.
			</div>

	== Remembering ==		__TOC__

			<div style="background-color: #000080; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Remembering</span> ==
	* '''Vector''' — A quantity having both magnitude and direction; an array of numbers.		* '''Vector''' — A quantity having both magnitude and direction; an array of numbers.
	* '''Matrix''' — A rectangular grid of numbers, arranged in rows and columns.		* '''Matrix''' — A rectangular grid of numbers, arranged in rows and columns.
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	* '''Linear Transformation''' — A function that moves space while keeping lines straight and the origin fixed.		* '''Linear Transformation''' — A function that moves space while keeping lines straight and the origin fixed.
	* '''System of Linear Equations''' — A set of two or more linear equations with the same variables.		* '''System of Linear Equations''' — A set of two or more linear equations with the same variables.
			</div>

	== Understanding ==		<div style="background-color: #006400; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Understanding</span> ==
	Linear Algebra is about '''Transforming Space'''.		Linear Algebra is about '''Transforming Space'''.

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	'''High-Dimensional Space''': We can only visualize 3D, but math doesn't care. Linear algebra works the same for a 1,000,000-dimensional vector. This is how AI works—a word is represented as a vector of 1536 numbers (Embeddings), and AI "calculates" the distance between words using the '''Dot Product'''.		'''High-Dimensional Space''': We can only visualize 3D, but math doesn't care. Linear algebra works the same for a 1,000,000-dimensional vector. This is how AI works—a word is represented as a vector of 1536 numbers (Embeddings), and AI "calculates" the distance between words using the '''Dot Product'''.
			</div>

	== Applying ==		<div style="background-color: #8B0000; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Applying</span> ==
	'''Modeling 'Image Transformations' (Scaling and Rotating):'''		'''Modeling 'Image Transformations' (Scaling and Rotating):'''
	<syntaxhighlight lang="python">		<syntaxhighlight lang="python">
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	: '''Deep Learning''' → Every layer of a neural network is just a "Matrix Multiplication" followed by a small non-linear tweak.		: '''Deep Learning''' → Every layer of a neural network is just a "Matrix Multiplication" followed by a small non-linear tweak.
	: '''Cryptography''' → Using matrix operations that are easy to do but "hard to undo" without a key (Lattice-based crypto).		: '''Cryptography''' → Using matrix operations that are easy to do but "hard to undo" without a key (Lattice-based crypto).
			</div>

	== Analyzing ==		<div style="background-color: #8B4500; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Analyzing</span> ==
	{\| class="wikitable"		{\| class="wikitable"
	\|+ Scalars vs. Vectors vs. Matrices		\|+ Scalars vs. Vectors vs. Matrices
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	'''The Concept of "Rank"''': The rank of a matrix tells you the "True Dimension" of the information inside it. If a 3x3 matrix has Rank 2, it means it has "collapsed" 3D space into a 2D flat plane. Analyzing the '''Null Space''' and '''Column Space''' of a matrix is how we find which equations in a system are actually redundant.		'''The Concept of "Rank"''': The rank of a matrix tells you the "True Dimension" of the information inside it. If a 3x3 matrix has Rank 2, it means it has "collapsed" 3D space into a 2D flat plane. Analyzing the '''Null Space''' and '''Column Space''' of a matrix is how we find which equations in a system are actually redundant.
			</div>

	== Evaluating ==		<div style="background-color: #483D8B; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Evaluating</span> ==
	Evaluating a linear model:		Evaluating a linear model:
	# '''Condition Number''': Is the matrix "well-behaved" or will a tiny error in input lead to a massive error in output?		# '''Condition Number''': Is the matrix "well-behaved" or will a tiny error in input lead to a massive error in output?
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	# '''Orthogonality''': Are the basis vectors "Perpendicular"? (Orthogonal bases are the "Gold Standard" for stability).		# '''Orthogonality''': Are the basis vectors "Perpendicular"? (Orthogonal bases are the "Gold Standard" for stability).
	# '''Computational Cost''': Is the matrix too big for a CPU? (This is why we use GPUs for AI).		# '''Computational Cost''': Is the matrix too big for a CPU? (This is why we use GPUs for AI).
			</div>

	== Creating ==		<div style="background-color: #2F4F4F; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Creating</span> ==
	Future Frontiers:		Future Frontiers:
	# '''Quantum Linear Algebra''': Using "Qubits" to perform matrix multiplications that are impossible for classical computers.		# '''Quantum Linear Algebra''': Using "Qubits" to perform matrix multiplications that are impossible for classical computers.
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	[[Category:Computer Science]]		[[Category:Computer Science]]
	[[Category:Physics]]		[[Category:Physics]]
			</div>

Wordpad: BloomWiki: Linear Algebra

2026-04-23T14:38:18Z

BloomWiki: Linear Algebra

← Older revision		Revision as of 14:38, 23 April 2026
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	== Evaluating ==		== Evaluating ==
	Evaluating a linear model: ~~(1)~~ '''Condition Number''': Is the matrix "well-behaved" or will a tiny error in input lead to a massive error in output? ~~(2)~~ '''Sparsity''': Is the matrix mostly zeros? (Symmetric/Sparse matrices are much faster for computers to solve). ~~(3)~~ '''Orthogonality''': Are the basis vectors "Perpendicular"? (Orthogonal bases are the "Gold Standard" for stability). ~~(4)~~ '''Computational Cost''': Is the matrix too big for a CPU? (This is why we use GPUs for AI).		Evaluating a linear model:
			# '''Condition Number''': Is the matrix "well-behaved" or will a tiny error in input lead to a massive error in output?
			# '''Sparsity''': Is the matrix mostly zeros? (Symmetric/Sparse matrices are much faster for computers to solve).
			# '''Orthogonality''': Are the basis vectors "Perpendicular"? (Orthogonal bases are the "Gold Standard" for stability).
			# '''Computational Cost''': Is the matrix too big for a CPU? (This is why we use GPUs for AI).

	== Creating ==		== Creating ==
	Future Frontiers: ~~(1)~~ '''Quantum Linear Algebra''': Using "Qubits" to perform matrix multiplications that are impossible for classical computers. ~~(2)~~ '''Geometric Deep Learning''': Applying linear algebra to data that lives on "Graphs" or "Manifolds" rather than flat grids. ~~(3)~~ '''Tensor Networks''': Going beyond matrices (2D) to Tensors (3D, 4D, 5D) to model complex physical systems like superconductors. ~~(4)~~ '''Fully Homomorphic Encryption''': Performing linear algebra on "Encrypted" data so a server can process your data without ever seeing it.		Future Frontiers:
			# '''Quantum Linear Algebra''': Using "Qubits" to perform matrix multiplications that are impossible for classical computers.
			# '''Geometric Deep Learning''': Applying linear algebra to data that lives on "Graphs" or "Manifolds" rather than flat grids.
			# '''Tensor Networks''': Going beyond matrices (2D) to Tensors (3D, 4D, 5D) to model complex physical systems like superconductors.
			# '''Fully Homomorphic Encryption''': Performing linear algebra on "Encrypted" data so a server can process your data without ever seeing it.

	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Computer Science]]		[[Category:Computer Science]]
	[[Category:Physics]]		[[Category:Physics]]

Wordpad: BloomWiki: Linear Algebra

2026-04-23T14:29:06Z

BloomWiki: Linear Algebra

← Older revision		Revision as of 14:29, 23 April 2026
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	* A specific matrix might '''Rotate''' everything 90 degrees.		* A specific matrix might '''Rotate''' everything 90 degrees.
	* Another might '''Scale''' everything to be 2x bigger.		* Another might '''Scale''' everything to be 2x bigger.
	* '''Matrix Multiplication''' is just "Stacking" these machines. Rotating then Scaling is one combined matrix.		* '''Matrix Multiplication''' is just "Stacking" these machines. Rotating ''then'' Scaling is one combined matrix.

	'''3. Eigenvectors (The Soul of the Matrix)''':		'''3. Eigenvectors (The Soul of the Matrix)''':

Wordpad: BloomWiki: Linear Algebra

2026-04-23T14:23:12Z

BloomWiki: Linear Algebra

← Older revision		Revision as of 14:23, 23 April 2026
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	{{BloomIntro}}		{{BloomIntro}}
	Linear Algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. If algebra is the study of single numbers, linear algebra is the study of Grids and Arrows. It is the "Engine Room" of modern technology. Every time you filter a photo on Instagram, search Google, or use an AI like ChatGPT, millions of linear algebra calculations are happening in the background. It allows us to solve thousands of equations simultaneously, making it the most practical and widely used math in the 21st century.		Linear Algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. If algebra is the study of single numbers, linear algebra is the study of '''Grids''' and '''Arrows'''. It is the "Engine Room" of modern technology. Every time you filter a photo on Instagram, search Google, or use an AI like ChatGPT, millions of linear algebra calculations are happening in the background. It allows us to solve thousands of equations simultaneously, making it the most practical and widely used math in the 21st century.

	== Remembering ==		== Remembering ==
Line 19:		Line 19:

	== Understanding ==		== Understanding ==
	Linear Algebra is about Transforming Space.		Linear Algebra is about '''Transforming Space'''.

	1. The Vector View: A vector $[3, 2]$ is a point in space. Linear algebra is the study of how to move, rotate, and stretch these points.		'''1. The Vector View''': A vector $[3, 2]$ is a point in space. Linear algebra is the study of how to move, rotate, and stretch these points.

	2. The Matrix as a Function: A matrix is a "Machine." You put a vector in, and a new vector comes out.		'''2. The Matrix as a Function''': A matrix is a "Machine." You put a vector in, and a new vector comes out.
	* A specific matrix might Rotate everything 90 degrees.		* A specific matrix might '''Rotate''' everything 90 degrees.
	* Another might Scale everything to be 2x bigger.		* Another might '''Scale''' everything to be 2x bigger.
	* Matrix Multiplication is just "Stacking" these machines. Rotating then Scaling is one combined matrix.		* '''Matrix Multiplication''' is just "Stacking" these machines. Rotating then Scaling is one combined matrix.

	3. Eigenvectors (The Soul of the Matrix):		'''3. Eigenvectors (The Soul of the Matrix)''':
	When you transform space, most vectors get knocked around and change direction. But for every transformation, there are special vectors—Eigenvectors—that stay pointing the same way. They are the "Axes of Rotation."		When you transform space, most vectors get knocked around and change direction. But for every transformation, there are special vectors—'''Eigenvectors'''—that stay pointing the same way. They are the "Axes of Rotation."
	* Google PageRank: Google treats the internet as a massive matrix. The "Most important websites" are actually the Eigenvectors of that matrix.		* '''Google PageRank''': Google treats the internet as a massive matrix. The "Most important websites" are actually the '''Eigenvectors''' of that matrix.

	High-Dimensional Space: We can only visualize 3D, but math doesn't care. Linear algebra works the same for a 1,000,000-dimensional vector. This is how AI works—a word is represented as a vector of 1536 numbers (Embeddings), and AI "calculates" the distance between words using the Dot Product.		'''High-Dimensional Space''': We can only visualize 3D, but math doesn't care. Linear algebra works the same for a 1,000,000-dimensional vector. This is how AI works—a word is represented as a vector of 1536 numbers (Embeddings), and AI "calculates" the distance between words using the '''Dot Product'''.

	== Applying ==		== Applying ==
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	\|}		\|}

	The Concept of "Rank": The rank of a matrix tells you the "True Dimension" of the information inside it. If a 3x3 matrix has Rank 2, it means it has "collapsed" 3D space into a 2D flat plane. Analyzing the Null Space and Column Space of a matrix is how we find which equations in a system are actually redundant.		'''The Concept of "Rank"''': The rank of a matrix tells you the "True Dimension" of the information inside it. If a 3x3 matrix has Rank 2, it means it has "collapsed" 3D space into a 2D flat plane. Analyzing the '''Null Space''' and '''Column Space''' of a matrix is how we find which equations in a system are actually redundant.

	== Evaluating ==		== Evaluating ==
	Evaluating a linear model: (1) Condition Number: Is the matrix "well-behaved" or will a tiny error in input lead to a massive error in output? (2) Sparsity: Is the matrix mostly zeros? (Symmetric/Sparse matrices are much faster for computers to solve). (3) Orthogonality: Are the basis vectors "Perpendicular"? (Orthogonal bases are the "Gold Standard" for stability). (4) Computational Cost: Is the matrix too big for a CPU? (This is why we use GPUs for AI).		Evaluating a linear model: (1) '''Condition Number''': Is the matrix "well-behaved" or will a tiny error in input lead to a massive error in output? (2) '''Sparsity''': Is the matrix mostly zeros? (Symmetric/Sparse matrices are much faster for computers to solve). (3) '''Orthogonality''': Are the basis vectors "Perpendicular"? (Orthogonal bases are the "Gold Standard" for stability). (4) '''Computational Cost''': Is the matrix too big for a CPU? (This is why we use GPUs for AI).

	== Creating ==		== Creating ==
	Future Frontiers: (1) Quantum Linear Algebra: Using "Qubits" to perform matrix multiplications that are impossible for classical computers. (2) Geometric Deep Learning: Applying linear algebra to data that lives on "Graphs" or "Manifolds" rather than flat grids. (3) Tensor Networks: Going beyond matrices (2D) to Tensors (3D, 4D, 5D) to model complex physical systems like superconductors. (4) Fully Homomorphic Encryption: Performing linear algebra on "Encrypted" data so a server can process your data without ever seeing it.		Future Frontiers: (1) '''Quantum Linear Algebra''': Using "Qubits" to perform matrix multiplications that are impossible for classical computers. (2) '''Geometric Deep Learning''': Applying linear algebra to data that lives on "Graphs" or "Manifolds" rather than flat grids. (3) '''Tensor Networks''': Going beyond matrices (2D) to Tensors (3D, 4D, 5D) to model complex physical systems like superconductors. (4) '''Fully Homomorphic Encryption''': Performing linear algebra on "Encrypted" data so a server can process your data without ever seeing it.

	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Computer Science]]		[[Category:Computer Science]]
	[[Category:Physics]]		[[Category:Physics]]

Wordpad: BloomWiki: Linear Algebra

2026-04-23T13:51:39Z

BloomWiki: Linear Algebra

New page

{{BloomIntro}}
Linear Algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. If algebra is the study of single numbers, linear algebra is the study of **Grids** and **Arrows**. It is the "Engine Room" of modern technology. Every time you filter a photo on Instagram, search Google, or use an AI like ChatGPT, millions of linear algebra calculations are happening in the background. It allows us to solve thousands of equations simultaneously, making it the most practical and widely used math in the 21st century.

== Remembering ==
* '''Vector''' — A quantity having both magnitude and direction; an array of numbers.
* '''Matrix''' — A rectangular grid of numbers, arranged in rows and columns.
* '''Scalar''' — A single number (as opposed to a vector or matrix).
* '''Dot Product''' — An operation that takes two vectors and returns a single number (measuring how much they "point" in the same direction).
* '''Matrix Multiplication''' — A way of combining two matrices to represent a complex transformation.
* '''Identity Matrix ($I$)''' — A square matrix with 1s on the diagonal and 0s elsewhere; the "1" of matrices.
* '''Inverse Matrix ($A^{-1}$)''' — A matrix that, when multiplied by $A$, result in the Identity matrix (used to "undo" a transformation).
* '''Determinant''' — A single number that describes the "Scale" of a transformation (if it's 0, the matrix has "collapsed" the space).
* '''Eigenvector''' — A vector whose direction does not change during a transformation.
* '''Eigenvalue''' — The factor by which an eigenvector is scaled during a transformation.
* '''Basis''' — A set of vectors that can be used to "build" any other vector in a space.
* '''Dimension''' — The number of vectors in a basis (e.g., 2D, 3D, or 1000D).
* '''Linear Transformation''' — A function that moves space while keeping lines straight and the origin fixed.
* '''System of Linear Equations''' — A set of two or more linear equations with the same variables.

== Understanding ==
Linear Algebra is about **Transforming Space**.

**1. The Vector View**: A vector $[3, 2]$ is a point in space. Linear algebra is the study of how to move, rotate, and stretch these points.

**2. The Matrix as a Function**: A matrix is a "Machine." You put a vector in, and a new vector comes out.
* A specific matrix might **Rotate** everything 90 degrees.
* Another might **Scale** everything to be 2x bigger.
* **Matrix Multiplication** is just "Stacking" these machines. Rotating *then* Scaling is one combined matrix.

**3. Eigenvectors (The Soul of the Matrix)**:
When you transform space, most vectors get knocked around and change direction. But for every transformation, there are special vectors—**Eigenvectors**—that stay pointing the same way. They are the "Axes of Rotation."
* **Google PageRank**: Google treats the internet as a massive matrix. The "Most important websites" are actually the **Eigenvectors** of that matrix.

**High-Dimensional Space**: We can only visualize 3D, but math doesn't care. Linear algebra works the same for a 1,000,000-dimensional vector. This is how AI works—a word is represented as a vector of 1536 numbers (Embeddings), and AI "calculates" the distance between words using the **Dot Product**.

== Applying ==
'''Modeling 'Image Transformations' (Scaling and Rotating):'''
<syntaxhighlight lang="python">
import numpy as np

def transform_point(x, y, scale_factor, rotation_deg):
"""
Shows how a Matrix 'moves' a point.
"""
theta = np.radians(rotation_deg)
# The Rotation Matrix
R = np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])

# The Scale Matrix
S = np.array([
[scale_factor, 0],
[0, scale_factor]
])

point = np.array([x, y])
# Apply Scale then Rotation
new_point = R @ S @ point
return new_point

# Move point [1, 0] by scaling 2x and rotating 90 deg
print(f"New Position: {transform_point(1, 0, 2, 90)}")
# This is how GPUs render 3D games and movies.
</syntaxhighlight>

; Linear Algebra in Tech
: '''Data Compression (SVD)''' → Breaking a giant matrix (like an image) into smaller ones to save space without losing detail.
: '''Principal Component Analysis (PCA)''' → Finding the most important "directions" in a massive dataset to simplify it.
: '''Deep Learning''' → Every layer of a neural network is just a "Matrix Multiplication" followed by a small non-linear tweak.
: '''Cryptography''' → Using matrix operations that are easy to do but "hard to undo" without a key (Lattice-based crypto).

== Analyzing ==
{| class="wikitable"
|+ Scalars vs. Vectors vs. Matrices
! Type !! Definition !! Practical Use
|-
| Scalar || A single number (5) || Price, Age, Temperature
|-
| Vector || A list of numbers [5, 2] || Position (x,y), Color (R,G,B), Word Meaning
|-
| Matrix || A grid of numbers || An Image, A Network, A Transformation
|}

**The Concept of "Rank"**: The rank of a matrix tells you the "True Dimension" of the information inside it. If a 3x3 matrix has Rank 2, it means it has "collapsed" 3D space into a 2D flat plane. Analyzing the **Null Space** and **Column Space** of a matrix is how we find which equations in a system are actually redundant.

== Evaluating ==
Evaluating a linear model: (1) **Condition Number**: Is the matrix "well-behaved" or will a tiny error in input lead to a massive error in output? (2) **Sparsity**: Is the matrix mostly zeros? (Symmetric/Sparse matrices are much faster for computers to solve). (3) **Orthogonality**: Are the basis vectors "Perpendicular"? (Orthogonal bases are the "Gold Standard" for stability). (4) **Computational Cost**: Is the matrix too big for a CPU? (This is why we use GPUs for AI).

== Creating ==
Future Frontiers: (1) **Quantum Linear Algebra**: Using "Qubits" to perform matrix multiplications that are impossible for classical computers. (2) **Geometric Deep Learning**: Applying linear algebra to data that lives on "Graphs" or "Manifolds" rather than flat grids. (3) **Tensor Networks**: Going beyond matrices (2D) to Tensors (3D, 4D, 5D) to model complex physical systems like superconductors. (4) **Fully Homomorphic Encryption**: Performing linear algebra on "Encrypted" data so a server can process your data without ever seeing it.

[[Category:Mathematics]]
[[Category:Computer Science]]
[[Category:Physics]]