Linear Algebra: Transformations, Spaces & Projections
How do matrices transform geometric space, and how do projections find optimal solutions in high dimensions?
How this lesson unfolds.
Linear algebra is the language of modern computation, graphics, and AI. Follow the grid as it stretches and rotates, see how projections solve unsolvable systems, and understand eigenvalues and SVD from pure visual intuition.
How does watch the grid transform work in linear algebra?
Think of watch the grid transform as a geometric action on space.
Watch The Grid Transform
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Visualizing watch the grid transform: see vectors, coordinates, and transformations operate directly in geometric space.
How does what happens to the axes work in linear algebra?
Think of what happens to the axes as a geometric action on space.
What Happens To The Axes
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Visualizing what happens to the axes: see vectors, coordinates, and transformations operate directly in geometric space.
How does columns tell directions work in linear algebra?
Think of columns tell directions as a geometric action on space.
Columns Tell Directions
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Visualizing columns tell directions: see vectors, coordinates, and transformations operate directly in geometric space.
How does column combination work in linear algebra?
Think of column combination as a geometric action on space.
Column Combination
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Visualizing column combination: see vectors, coordinates, and transformations operate directly in geometric space.
How does rows calculate columns build work in linear algebra?
Think of rows calculate columns build as a geometric action on space.
Rows Calculate Columns Build
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Visualizing rows calculate columns build: see vectors, coordinates, and transformations operate directly in geometric space.
How does three inputs two outputs work in linear algebra?
Think of three inputs two outputs as a geometric action on space.
Three Inputs Two Outputs
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Visualizing three inputs two outputs: see vectors, coordinates, and transformations operate directly in geometric space.
How does order changes answer work in linear algebra?
Think of order changes answer as a geometric action on space.
Order Changes Answer
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Visualizing order changes answer: see vectors, coordinates, and transformations operate directly in geometric space.
How does regroup dont reorder work in linear algebra?
Think of regroup dont reorder as a geometric action on space.
Regroup Dont Reorder
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Visualizing regroup dont reorder: see vectors, coordinates, and transformations operate directly in geometric space.
How does transpose swaps work in linear algebra?
Think of transpose swaps as a geometric action on space.
Transpose Swaps
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Visualizing transpose swaps: see vectors, coordinates, and transformations operate directly in geometric space.
How does can matrix reach b work in linear algebra?
Think of can matrix reach b as a geometric action on space.
Can Matrix Reach B
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Visualizing can matrix reach b: see vectors, coordinates, and transformations operate directly in geometric space.
How does image reachable outputs work in linear algebra?
Think of image reachable outputs as a geometric action on space.
Image Reachable Outputs
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Visualizing image reachable outputs: see vectors, coordinates, and transformations operate directly in geometric space.
How does one vector infinite multiples work in linear algebra?
Think of one vector infinite multiples as a geometric action on space.
One Vector Infinite Multiples
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Visualizing one vector infinite multiples: see vectors, coordinates, and transformations operate directly in geometric space.
How does two directions one plane work in linear algebra?
Think of two directions one plane as a geometric action on space.
Two Directions One Plane
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Visualizing two directions one plane: see vectors, coordinates, and transformations operate directly in geometric space.
How does two vectors one direction work in linear algebra?
Think of two vectors one direction as a geometric action on space.
Two Vectors One Direction
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Visualizing two vectors one direction: see vectors, coordinates, and transformations operate directly in geometric space.
How does basis enough not extra work in linear algebra?
Think of basis enough not extra as a geometric action on space.
Basis Enough Not Extra
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Visualizing basis enough not extra: see vectors, coordinates, and transformations operate directly in geometric space.
How does rank counts output dimensions work in linear algebra?
Think of rank counts output dimensions as a geometric action on space.
Rank Counts Output Dimensions
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Visualizing rank counts output dimensions: see vectors, coordinates, and transformations operate directly in geometric space.
How does plane becomes line work in linear algebra?
Think of plane becomes line as a geometric action on space.
Plane Becomes Line
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Visualizing plane becomes line: see vectors, coordinates, and transformations operate directly in geometric space.
How does two inputs one output work in linear algebra?
Think of two inputs one output as a geometric action on space.
Two Inputs One Output
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Visualizing two inputs one output: see vectors, coordinates, and transformations operate directly in geometric space.
How does kernel what disappears work in linear algebra?
Think of kernel what disappears as a geometric action on space.
Kernel What Disappears
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Visualizing kernel what disappears: see vectors, coordinates, and transformations operate directly in geometric space.
How does invisible movement same result work in linear algebra?
Think of invisible movement same result as a geometric action on space.
Invisible Movement Same Result
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Visualizing invisible movement same result: see vectors, coordinates, and transformations operate directly in geometric space.
How does reachable not unique work in linear algebra?
Think of reachable not unique as a geometric action on space.
Reachable Not Unique
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Visualizing reachable not unique: see vectors, coordinates, and transformations operate directly in geometric space.
How does not every target reachable work in linear algebra?
Think of not every target reachable as a geometric action on space.
Not Every Target Reachable
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Visualizing not every target reachable: see vectors, coordinates, and transformations operate directly in geometric space.
How does existence vs uniqueness work in linear algebra?
Think of existence vs uniqueness as a geometric action on space.
Existence Vs Uniqueness
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Visualizing existence vs uniqueness: see vectors, coordinates, and transformations operate directly in geometric space.
How does three possible outcomes work in linear algebra?
Think of three possible outcomes as a geometric action on space.
Three Possible Outcomes
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Visualizing three possible outcomes: see vectors, coordinates, and transformations operate directly in geometric space.
How does different equations same solution work in linear algebra?
Think of different equations same solution as a geometric action on space.
Different Equations Same Solution
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Visualizing different equations same solution: see vectors, coordinates, and transformations operate directly in geometric space.
How does no exact hit find nearest work in linear algebra?
Think of no exact hit find nearest as a geometric action on space.
No Exact Hit Find Nearest
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Visualizing no exact hit find nearest: see vectors, coordinates, and transformations operate directly in geometric space.
How does projection finds closest point work in linear algebra?
Think of projection finds closest point as a geometric action on space.
Projection Finds Closest Point
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Visualizing projection finds closest point: see vectors, coordinates, and transformations operate directly in geometric space.
How does original projection residual work in linear algebra?
Think of original projection residual as a geometric action on space.
Original Projection Residual
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Visualizing original projection residual: see vectors, coordinates, and transformations operate directly in geometric space.
How does why perpendicular is best work in linear algebra?
Think of why perpendicular is best as a geometric action on space.
Why Perpendicular Is Best
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Visualizing why perpendicular is best: see vectors, coordinates, and transformations operate directly in geometric space.
How does one vector two parts work in linear algebra?
Think of one vector two parts as a geometric action on space.
One Vector Two Parts
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Visualizing one vector two parts: see vectors, coordinates, and transformations operate directly in geometric space.
How does cosine calculates shadow work in linear algebra?
Think of cosine calculates shadow as a geometric action on space.
Cosine Calculates Shadow
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Visualizing cosine calculates shadow: see vectors, coordinates, and transformations operate directly in geometric space.
How does projection nonunit vector work in linear algebra?
Think of projection nonunit vector as a geometric action on space.
Projection Nonunit Vector
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Visualizing projection nonunit vector: see vectors, coordinates, and transformations operate directly in geometric space.
How does longer u same shadow work in linear algebra?
Think of longer u same shadow as a geometric action on space.
Longer U Same Shadow
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Visualizing longer u same shadow: see vectors, coordinates, and transformations operate directly in geometric space.
How does residual perpendicular work in linear algebra?
Think of residual perpendicular as a geometric action on space.
Residual Perpendicular
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Visualizing residual perpendicular: see vectors, coordinates, and transformations operate directly in geometric space.
How does complementary spaces work in linear algebra?
Think of complementary spaces as a geometric action on space.
Complementary Spaces
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Visualizing complementary spaces: see vectors, coordinates, and transformations operate directly in geometric space.
How does projection works 3d work in linear algebra?
Think of projection works 3d as a geometric action on space.
Projection Works 3D
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Visualizing projection works 3d: see vectors, coordinates, and transformations operate directly in geometric space.
How does matrix casts shadows work in linear algebra?
Think of matrix casts shadows as a geometric action on space.
Matrix Casts Shadows
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Visualizing matrix casts shadows: see vectors, coordinates, and transformations operate directly in geometric space.
How does projection once enough work in linear algebra?
Think of projection once enough as a geometric action on space.
Projection Once Enough
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Visualizing projection once enough: see vectors, coordinates, and transformations operate directly in geometric space.
How does projection respects addition work in linear algebra?
Think of projection respects addition as a geometric action on space.
Projection Respects Addition
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Visualizing projection respects addition: see vectors, coordinates, and transformations operate directly in geometric space.
How does best fit projection work in linear algebra?
Think of best fit projection as a geometric action on space.
Best Fit Projection
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Visualizing best fit projection: see vectors, coordinates, and transformations operate directly in geometric space.
How does smallest residual wins work in linear algebra?
Think of smallest residual wins as a geometric action on space.
Smallest Residual Wins
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Visualizing smallest residual wins: see vectors, coordinates, and transformations operate directly in geometric space.
How does best error perpendicular work in linear algebra?
Think of best error perpendicular as a geometric action on space.
Best Error Perpendicular
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Visualizing best error perpendicular: see vectors, coordinates, and transformations operate directly in geometric space.
How does normal equation fit work in linear algebra?
Think of normal equation fit as a geometric action on space.
Normal Equation Fit
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Visualizing normal equation fit: see vectors, coordinates, and transformations operate directly in geometric space.
How does inverse undoes transformation work in linear algebra?
Think of inverse undoes transformation as a geometric action on space.
Inverse Undoes Transformation
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Visualizing inverse undoes transformation: see vectors, coordinates, and transformations operate directly in geometric space.
How does pseudoinverse best answer work in linear algebra?
Think of pseudoinverse best answer as a geometric action on space.
Pseudoinverse Best Answer
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Visualizing pseudoinverse best answer: see vectors, coordinates, and transformations operate directly in geometric space.
How does project using model matrix work in linear algebra?
Think of project using model matrix as a geometric action on space.
Project Using Model Matrix
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Visualizing project using model matrix: see vectors, coordinates, and transformations operate directly in geometric space.
How does sound vector work in linear algebra?
Think of sound vector as a geometric action on space.
Sound Vector
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Visualizing sound vector: see vectors, coordinates, and transformations operate directly in geometric space.
How does waves combine like vectors work in linear algebra?
Think of waves combine like vectors as a geometric action on space.
Waves Combine Like Vectors
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Visualizing waves combine like vectors: see vectors, coordinates, and transformations operate directly in geometric space.
How does signal fitting projection work in linear algebra?
Think of signal fitting projection as a geometric action on space.
Signal Fitting Projection
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Visualizing signal fitting projection: see vectors, coordinates, and transformations operate directly in geometric space.
How does now you can see mathematics work in linear algebra?
Think of now you can see mathematics as a geometric action on space.
Now You Can See Mathematics
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Visualizing now you can see mathematics: see vectors, coordinates, and transformations operate directly in geometric space.
What is Equality Connects Two Representations?
Understand Equality Connects Two Representations geometrically before computing algebraic coordinates.
xᵀy = Σᵢ xᵢyᵢ
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r.
What is Symbols Tell You the Object?
Understand Symbols Tell You the Object geometrically before computing algebraic coordinates.
a∈R
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What is Shape Is the First Error Check?
Understand Shape Is the First Error Check geometrically before computing algebraic coordinates.
A∈Rᵐˣⁿ, x∈Rⁿ ⇒ Ax∈Rᵐ
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What is Sigma Is a Loop in Disguise?
Understand Sigma Is a Loop in Disguise geometrically before computing algebraic coordinates.
Σᵢ₌₁⁴ i=10
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What is Which Axis Are You Summing Over??
Understand Which Axis Are You Summing Over? geometrically before computing algebraic coordinates.
Σ_d w_dx_d
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What is Same Symbol, Different Job?
Understand Same Symbol, Different Job geometrically before computing algebraic coordinates.
λ
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What is Bars Are Not All the Same?
Understand Bars Are Not All the Same geometrically before computing algebraic coordinates.
Bars Are Not All the Same
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What is An Inner Product Chooses Geometry?
Understand An Inner Product Chooses Geometry geometrically before computing algebraic coordinates.
⟨x,y⟩_M=xᵀMy
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What is Change the Metric, Change the Angle?
Understand Change the Metric, Change the Angle geometrically before computing algebraic coordinates.
M=diag(4,1)
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What is Matrices Need a Norm Too?
Understand Matrices Need a Norm Too geometrically before computing algebraic coordinates.
‖A‖_F=√Σᵢⱼaᵢⱼ²
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What is Reconstruction Error Is Matrix Distance?
Understand Reconstruction Error Is Matrix Distance geometrically before computing algebraic coordinates.
‖X-X̂‖_F
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What is Put the System into One Table?
Understand Put the System into One Table geometrically before computing algebraic coordinates.
Put the System into One Tableb]` for `x₁+2x₂=5`, `3x₁+4x₂=11`.
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What is Three Moves Preserve Solutions?
Understand Three Moves Preserve Solutions geometrically before computing algebraic coordinates.
Three Moves Preserve Solutions
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What is Elimination Reveals the Pivots?
Understand Elimination Reveals the Pivots geometrically before computing algebraic coordinates.
(x₁,x₂)=(1,2)
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What is Homogeneous Means Aim at Zero?
Understand Homogeneous Means Aim at Zero geometrically before computing algebraic coordinates.
Ax=0
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What is Pivot Variables Are Forced?
Understand Pivot Variables Are Forced geometrically before computing algebraic coordinates.
Pivot Variables Are Forced
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What is Free Variables Draw the Kernel?
Understand Free Variables Draw the Kernel geometrically before computing algebraic coordinates.
x₁+2x₂-x₃=0
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What is Rank Plus Nullity Equals Input Dimension?
Understand Rank Plus Nullity Equals Input Dimension geometrically before computing algebraic coordinates.
rank(A)+nullity(A)=n
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What is Every Input Direction Has a Fate?
Understand Every Input Direction Has a Fate geometrically before computing algebraic coordinates.
Every Input Direction Has a Fate
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What is Image Equals Column Space—for Linear Maps?
Understand Image Equals Column Space—for Linear Maps geometrically before computing algebraic coordinates.
Im(A)=Col(A)
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What is Nonlinear Images Need Not Be Subspaces?
Understand Nonlinear Images Need Not Be Subspaces geometrically before computing algebraic coordinates.
f(x)=x²
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What is Four Subspaces Organize a Matrix?
Understand Four Subspaces Organize a Matrix geometrically before computing algebraic coordinates.
Col(A)
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What is Null Spaces Are Orthogonal Complements?
Understand Null Spaces Are Orthogonal Complements geometrically before computing algebraic coordinates.
Null(A)=Row(A)⊥
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What is One Matrix, Four Spaces, One Balance?
Understand One Matrix, Four Spaces, One Balance geometrically before computing algebraic coordinates.
r,n-r,r,m-r
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What is Linear Does Not Mean Straight?
Understand Linear Does Not Mean Straight geometrically before computing algebraic coordinates.
Linear Does Not Mean Straight
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What is A Set Must Pass Two Tests?
Understand A Set Must Pass Two Tests geometrically before computing algebraic coordinates.
A Set Must Pass Two Tests
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What is The Unit Circle Fails?
Understand The Unit Circle Fails geometrically before computing algebraic coordinates.
x²+y²=1
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What is Choose the Waves You Allow?
Understand Choose the Waves You Allow geometrically before computing algebraic coordinates.
U=span{sin(ωt),cos(ωt)}
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What is Four Samples Become One Target?
Understand Four Samples Become One Target geometrically before computing algebraic coordinates.
y=[1.0,2.2,2.9,1.1]ᵀ
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What is Two Basis Signals Become a Matrix?
Understand Two Basis Signals Become a Matrix geometrically before computing algebraic coordinates.
X=[[1,0],[2,1],[1,2],[0,1]]
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What is Fit Both Waves at Once?
Understand Fit Both Waves at Once geometrically before computing algebraic coordinates.
min_θ‖Xθ-y‖²
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What is Normal Equations Solve the Signal?
Understand Normal Equations Solve the Signal geometrically before computing algebraic coordinates.
XᵀX=[[6,4],[4,6]]Logic: orthogonality yields `θ=(0.67,1.07)`; next verifies.
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What is The Residual Is Silent to the Basis?
Understand The Residual Is Silent to the Basis geometrically before computing algebraic coordinates.
ŷ=[0.67,2.41,2.81,1.07]ᵀ
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What is One Projector Fits Every Signal?
Understand One Projector Fits Every Signal geometrically before computing algebraic coordinates.
P=X(XᵀX)⁻¹Xᵀ
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What is One Formula, Many Norms?
Understand One Formula, Many Norms geometrically before computing algebraic coordinates.
One Formula, Many Norms
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What is Unit Balls Reveal the Rule?
Understand Unit Balls Reveal the Rule geometrically before computing algebraic coordinates.
Unit Balls Reveal the RuleShared axes and equal-radius contours.
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What is Why L1 Finds Corners?
Understand Why L1 Finds Corners geometrically before computing algebraic coordinates.
Why L1 Finds Corners
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What is Why L2 Shrinks Smoothly?
Understand Why L2 Shrinks Smoothly geometrically before computing algebraic coordinates.
Why L2 Shrinks Smoothly
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What is L∞ Guards the Worst Coordinate?
Understand L∞ Guards the Worst Coordinate geometrically before computing algebraic coordinates.
L∞ Guards the Worst Coordinate
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What is Metric Choice Changes the Neighbor?
Understand Metric Choice Changes the Neighbor geometrically before computing algebraic coordinates.
Metric Choice Changes the Neighbor
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What is Some Directions Keep Their Line?
Understand Some Directions Keep Their Line geometrically before computing algebraic coordinates.
Av=λv
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What is Eigenvalue Says How Much?
Understand Eigenvalue Says How Much geometrically before computing algebraic coordinates.
Eigenvalue Says How Much
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What is A Matrix Can Reveal Its Own Axes?
Understand A Matrix Can Reveal Its Own Axes geometrically before computing algebraic coordinates.
A Matrix Can Reveal Its Own Axes
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What is Covariance Draws a Data Ellipse?
Understand Covariance Draws a Data Ellipse geometrically before computing algebraic coordinates.
Covariance Draws a Data Ellipse
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What is Variance Depends on Direction?
Understand Variance Depends on Direction geometrically before computing algebraic coordinates.
wᵀΣw
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What is PCA Chooses Maximum Spread?
Understand PCA Chooses Maximum Spread geometrically before computing algebraic coordinates.
max_{‖w‖=1} wᵀΣw
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What is The Top Eigenvector Wins?
Understand The Top Eigenvector Wins geometrically before computing algebraic coordinates.
The Top Eigenvector Wins
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What is Projection Reduces Dimension?
Understand Projection Reduces Dimension geometrically before computing algebraic coordinates.
Projection Reduces Dimension
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What is Every Matrix Has Three Stages?
Understand Every Matrix Has Three Stages geometrically before computing algebraic coordinates.
A=UΣVᵀ
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What is Singular Values Rank Importance?
Understand Singular Values Rank Importance geometrically before computing algebraic coordinates.
Singular Values Rank Importance
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