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Linear Algebra
X1 = [ 2 1 2], X2 = [ 1 -1 -2 ] , X3 = [ 1, 1, 1]
find the dimension and a set of basis vector of V
find the dimension and a set of basis vector of V
Linear Algebra
linear equations using gauss elimination method x+y+z=2 2x-3y=5 -3x+2z=1
Linear Algebra
D. Solving a System of Linear Equation.
Direction: Use an inverse matrix to solve each system of linear equation.
1. (a) x+2y=-1
x-2y=3
(b) x+2y=10
x-2y=-6
2. (a) 2x-y=-3
2x+y=7
(b) 2x-y=-1
2x+y=-3
3. (a) x+2y+z=2
x+2y-z=4
x-2y+z=-2
(b) x+2y+z=1
x+2y-z=3
x-2y+z=-3
E. Let A, B and C be:
A = [1 2 -3; 0 1 2; -1 2 0]
B = [-1 2 0; 0 1 2; 1 2 -3]
C = [0 4 -3; 0 1 2; -1 2 0]
1. Find an elementary matrix E such that EA=B
2. Find an elementary matrix E such that EC=A
3. Find an elementary matrix E such that EB=A
F. Find the LU Factorization of the matrix.
4. A = [1 0; -2 1]
5. B = [-2 1; -6 4]
6. C = [3 0 1; 6 1 1; -3 1 0]
Direction: Use an inverse matrix to solve each system of linear equation.
1. (a) x+2y=-1
x-2y=3
(b) x+2y=10
x-2y=-6
2. (a) 2x-y=-3
2x+y=7
(b) 2x-y=-1
2x+y=-3
3. (a) x+2y+z=2
x+2y-z=4
x-2y+z=-2
(b) x+2y+z=1
x+2y-z=3
x-2y+z=-3
E. Let A, B and C be:
A = [1 2 -3; 0 1 2; -1 2 0]
B = [-1 2 0; 0 1 2; 1 2 -3]
C = [0 4 -3; 0 1 2; -1 2 0]
1. Find an elementary matrix E such that EA=B
2. Find an elementary matrix E such that EC=A
3. Find an elementary matrix E such that EB=A
F. Find the LU Factorization of the matrix.
4. A = [1 0; -2 1]
5. B = [-2 1; -6 4]
6. C = [3 0 1; 6 1 1; -3 1 0]
Linear Algebra
A. Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination.
1.-x+y+2z=1
2x+3y+z=-2
5x+4y+2z=4
2. 2x+3y+z=10
2x-3y-3z=22
4x-2y+3z=-2
B. Finding the Inverse of a Matrix.
Find the inverse of the matrix.
1. [2 0; 0 3]
2. [-1 1; 3 -3]
3. [1 2; 3 7]
C. Finding the inverse of the Square of a Matrix.
Direction: Compute A^-2.
1. A = [0 -2; -1 3]
2. A = [2 7; -5 6]
3. A = [-2 0 0; 0 1 0; 0 0 3]
1.-x+y+2z=1
2x+3y+z=-2
5x+4y+2z=4
2. 2x+3y+z=10
2x-3y-3z=22
4x-2y+3z=-2
B. Finding the Inverse of a Matrix.
Find the inverse of the matrix.
1. [2 0; 0 3]
2. [-1 1; 3 -3]
3. [1 2; 3 7]
C. Finding the inverse of the Square of a Matrix.
Direction: Compute A^-2.
1. A = [0 -2; -1 3]
2. A = [2 7; -5 6]
3. A = [-2 0 0; 0 1 0; 0 0 3]
Linear Algebra
Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases
B1 = {(1, 0, 0),(0, 1, 0),(0, 0, 1)}, B2 = {(1, 0, 0),(0, 1, 2),(0, 2, 1)}. If
Q(X) = 2x1^2+ 2x1x2 − 2x2x3 +x2^2+x3^2, find the representation of Q in terms of
(y1,y2,y3).
B1 = {(1, 0, 0),(0, 1, 0),(0, 0, 1)}, B2 = {(1, 0, 0),(0, 1, 2),(0, 2, 1)}. If
Q(X) = 2x1^2+ 2x1x2 − 2x2x3 +x2^2+x3^2, find the representation of Q in terms of
(y1,y2,y3).
Linear Algebra
Find the nature of the conic 2x^2 + xy− y^2− 6 = 0.
Linear Algebra
Show that an orthogonal map of the plane is either a reflection, or a rotation.
Linear Algebra
Find a subspace U of P5 such that W ⊕U = P5. What is the dimension of U?
Linear Algebra
Apply the Gram-Schmidt diagonalisation process to find an orthonormal basis for the
subspace of C^4 generated by the vectors
{(1,−i,0,1),(−1,0, i,0),(−i,0,1,−1)}
subspace of C^4 generated by the vectors
{(1,−i,0,1),(−1,0, i,0),(−i,0,1,−1)}
Linear Algebra
Let T : P1 → P2 be defined by
T(a+bx) = b+ax+(a−b)x^2.
Check that T is a linear transformation. Find the matrix of the transformation with
respect to the ordered bases B1 = {1,x} and B2 = {x^2,x^2+x,x^2+x+1}. Find the
kernel of T . Further check that the range of T is
{a+bx+cx^2 ∈ P2 | a+c = b}.
T(a+bx) = b+ax+(a−b)x^2.
Check that T is a linear transformation. Find the matrix of the transformation with
respect to the ordered bases B1 = {1,x} and B2 = {x^2,x^2+x,x^2+x+1}. Find the
kernel of T . Further check that the range of T is
{a+bx+cx^2 ∈ P2 | a+c = b}.
