Linear Algebra | Original
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Singular Value Decomposition can be used for image compression.
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The determinant of a matrix indicates how much the area changes when the matrix is applied to a shape.
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The eigenvalues are defined by the equation \[ \det(\lambda I - A) = 0 \].
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NumPy’s
linalgpackage can be used to study and solve problems in linear algebra. -
Use linear-algebra transforms in n-dimensional space to understand matrices.
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A matrix is a group of numbers arranged in rows and columns.
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The equation \[ Ax = B \] has a solution when \[ \text{rank}(A) = \text{rank}([A B]) \]. - The rank of a matrix is the number of linearly independent rows (or columns) in its echelon form.
What is the use of the rank concept? It tells us how many rows retain information during a linear transformation. In contrast, the nullity (dimension of the null space) counts how many rows are mapped to zero.
What is the difference between a matrix and a plain array of numbers? An array has no inherent direction, whereas a matrix does.
What is the difference between a matrix and a vector? A vector is a one-dimensional column of numbers and has a direction.
Matrix multiplication follows a fixed rule: the number of columns in the first matrix must equal the number of rows in the second matrix. A 2×3 matrix multiplied by a 3×4 matrix yields a 2×4 matrix. In the resulting matrix, the value at position (i, j) is the dot product of the i-th row of matrix A and the j-th column of matrix B.
The determinant of a matrix carries meaning: if it is not equal to 0, the matrix is invertible.