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These are the notes of Exam of Linear Algebra which includes Initial Value Problem, General Solution, Erential Equation, Origin Parallel, Line, Vector Space, Dimension etc. Key important points are: Determinant, Matrix, Inverse, Compute the Inverse, Linear System, Linear Transformations, Volume, Determine, Basis, Column Spaces
Typology: Exams
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Find the determinant of the matrix below. Specify whether the matrix has an inverse without trying to compute the inverse. (^)
Solve the linear system using Cramer’s Rule:
2 x 1 + 3 x 2 − x 3 = 2 3 x 1 − 2 x 2 + x 3 = − 1 − 5 x 1 − 4 x 2 + 2 x 3 = 3
Let v ⃗ 1 = (1 , 1 , 1), v ⃗ 2 = (1 , 2 , 3) and v⃗ (^) 3 = (1 , 1 , 2). a. Show that the vectors above are linearly independent. b. Find the unique scalars(weights) c 2 , c 2 , c 3 such that v⃗ = (2 , 1 , 3) can be written as v ⃗ = cv 1 ⃗ 2 + cv 2 ⃗ 2 + cv 3 ⃗ 3.
Define a linear transformation T : P 2 Ï P 2 by T (p( x )) = p ′ ( x ). a. Describe the range of T b. Find dim ( R ( T )). c. Find dim ( N ( T )).
