# Tag Archives: Linear Algebra

## Superposition Principle of Linear Algebra

Superposition Principle If is a solution to , and is a solution to , then is a solution to . Proof: Properties of Matrix Transpose 1) 2) 3) 4)

## Characteristic Polynomial, Eigenvalues, Eigenvectors

Characteristic Polynomial, Eigenspace The solution space of is called the eigenspace of associated with the eigenvalue . The eigenspace is denoted by . Sum/Product of Eigenvalues – The sum of all eigenvalues of (including repeated eigenvalues) is the same as … Continue reading

## Finding Least Squares Solution Review and Others

Rotation Matrix The rotation matrix rotates points in the -plane counterclockwise through an angle about the origin. For example rotating the vector 45 degrees counterclockwise gives us: Finding Least Squares Solution Given (inconsistent system), solve instead to get a least … Continue reading

## Dot Product and Span Summary

Dot Product – – Span Subspaces is a subspace of if 1) for some vectors . 2) satisfies the closure properties: (i) for all , we must have . (ii) for all and , we must have . 3) is … Continue reading

## Gaussian Elimination Summary

Row echelon form (REF) For each non-zero row, the leading entry is to the right of the leading entry of the row above. E.g. Note that the leading entry 9 of the second row is to the right of the … Continue reading

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## Vector Subspace Question (GRE 0568 Q3)

This is an interesting question on vector subspaces (a topic from linear algebra): Question: If V and W are 2-dimensional subspaces of , what are the possible dimensions of the subspace ? (A) 1 only (B) 2 only (C) 0 … Continue reading