Linear Algebra
Essential formulas and concepts in linear algebra
Eigenvalue Equation
\(\det(A - \lambda I) = 0\)
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Mathematical concept visualization
Dot Product
\(\mathbf{u}\cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z = |\mathbf{u}|\;|\mathbf{v}|\cos\theta\)
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Description/Usage:
Produces a scalar equal to the sum of component-wise products. It measures how much one vector extends in the direction of another. If the dot product is zero, vectors are perpendicular (orthogonal).
Mathematical concept visualization
Cross Product (3D vectors)
\(\mathbf{u} \times \mathbf{v} = (u_y v_z - u_z v_y,\; u_z v_x - u_x v_z,\; u_x v_y - u_y v_x)\)
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Mathematical concept visualization