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**Determinant:** A scalar value from a square matrix. It indicates volume scaling & invertibility.
Ex: `det([a,b; c,d]) = ad-bc`.
Pro-Tip: If `det(A) = 0`, matrix A is singular (not invertible)!
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Alright, future engineers!
**Linear Independence:** Vectors are LI if none can be formed by combining the others.
Ex: `v1, v2` are LI if `c1v1+c2v2=0` means `c1=c2=0`.
Pro-Tip: LI vectors add new information & are key for a basis!
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Alright, future engineers!
**Linear Combination:** A sum of vectors, each multiplied by a scalar.
Ex: `c1*v1 + c2*v2`.
Pro-Tip: Crucial for understanding span & basis! Think of mixing vectors.
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**Matrix Multiplication:** Combining two matrices to get a new one, representing composed transformations.
Ex: `(2x3) * (3x2)` gives a `(2x2)` matrix.
Pro-Tip: Inner dimensions MUST match (n in m x n * n x p)!
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**Dot Product:** A scalar value representing how much two vectors point in the same direction, or the projection of one onto another.
Ex: `[1,2] . [3,4] = 1*3 + 2*4 = 11`.
Pro-Tip: If their dot product is zero, the vectors are perpendicular! Essential for geometric understanding.
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Alright, future engineers!
**Dot Product (Scalar Product):** Multiplies two vectors to give a *scalar* measuring their directional alignment.
Ex: `[1,2] . [3,4] = 1*3 + 2*4 = 11`.
Pro-Tip: If `a . b = 0`, the vectors are perfectly orthogonal (perpendicular)! This is KEY for geometry & projections.
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Matrix Addition: Sums matrices by adding corresponding elements. Dimensions MUST match!
Ex: `[[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]]`
Pro-Tip: If dimensions differ, addition is undefined! Crucial for compatibility in systems.
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Alright, future engineers!
**Dot Product:** Measures how much two vectors align, producing a scalar.
Ex: `[1,2] · [3,4] = (1*3)+(2*4) = 11`.
Pro-Tip: If it's zero, vectors are orthogonal! Essential for projections & angles.
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