Linear Algebra Linear Dependence of Vector Spaces

Question : Prove that ( 1 , 3 , 2 ) , ( 1 , – 7 , – 8 ) , ( 2 , 1 , – 1 ) of V3( R) is linearly independent.

Max Z = 5x1 + 6x2 s.t. 17x1 + 8x2 < = 136 3x1 + 4x2 < = 36 x1, x2 > = 0

What is the optimal solution? Z = ?

Finding Basis and Dimension of the Subspace generated by given vectors. For complete description of the specific problems please see the problems.

Question (1) Find a basis and dimension of the subspace W of R4 generated by the vectors ( 1 , – 4 , – 2 , 1 ) , ( 1 , – 3 , – 1 , 2 ) , ( 3 , – 8 , – 2 , 7 ) . Extend it to find the basis of R4 . Question (2) Determine a basis and the dimensions of the Subspace of M2(R) generated by the 2 by 2 Matrices [ 2 -10 ] , [ 3 3 ] , [ 2 -4 ] , [ 2 -14 ] [ -8 4 ] [ -3 15] [ -5 7] [ -10 2 ] NOTE : For the full description of the question in mathematical font and format, please download the attached question file.

Finding Basis and Dimension of the Subspace generated by given vectors. For complete description of the specific problems please see the problems.

Question (1) Find a basis and dimension of the subspace W of R4 generated by the vectors ( 1 , – 4 , – 2 , 1 ) , ( 1 , – 3 , – 1 , 2 ) , ( 3 , – 8 , – 2 , 7 ) . Extend it to find the basis of R4 Question (2) Determine a basis and the dimensions of the Subspace of M2(R) generated by the 2 by 2 Matrices [ 2 -10 ] , [ 3 3 ] , [ 2 -4 ] , [ 2 -14 ] [ -8 4 ] [ -3 15] [ -5 7] [ -10 2 ] NOTE : For the full description of the question in mathematical font and format, please download the attached question file.

Linear Operators Solving Equations Finding Annihilator of a Space

Question (4) Solve the equations over R X1 + 2X2 – 3X3 + 4X4 = 0 X1 + 3X2 – X3 = 0 6X1 + X3 + 2X4 = 0 Question(5) If F = R find Annihilator A(W) of the space W spanned by (2 , 4 , 6 ) , ( 1 , 6 , 2 ). ( Note : Here F is the field and R represents the set of Real Numbers) See attached file for full problem description.