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Linear Algebra
Let V be a vector space over a field F and let T : V → V be a linear operator. Show
T(W) ⊂ W for any subspace W of V if and only if there is a λ ∈ F such that Tv = λv
for all v ∈ V .
T(W) ⊂ W for any subspace W of V if and only if there is a λ ∈ F such that Tv = λv
for all v ∈ V .
Linear Algebra
Find the orthogonal canonical reduction of the quadratic form
−x2 + y2 + z2− 6xy− 6xz+ 2yz . Also, find its principal axes, rank and signature of the
quadratic form.
−x2 + y2 + z2− 6xy− 6xz+ 2yz . Also, find its principal axes, rank and signature of the
quadratic form.
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) = 2x21 +2x1x2 −2x2x3 +x22 +x23, 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) = 2x21 +2x1x2 −2x2x3 +x22 +x23, find the representation of Q in terms of (y1, y2, y3).
Linear Algebra
Find the polynomial of degree 4 whose graph goes through the points (−3,−296),(−2,−60),(−2,−60), (0,4), (2,4) and (3,−110).
Linear Algebra
Find the quadratic polynomial whose graph goes through the points (−2,10),(−2,10), (0,6),(0,6), and (1,10)(1,10).
Linear Algebra
Solve the systems using the Gaussian elimination method.
x+2y+z=4
2x-y-3z=2
x-8y-9z=-8
5x+5y =14
x+2y+z=4
2x-y-3z=2
x-8y-9z=-8
5x+5y =14
Linear Algebra
4. a) The Gauss elimination method is used to solve the system of equations
6 x 4x x 1 + 2 + α 3 =
3 2x x 2 x 1 − 2 + α 3 =
5 x 3x x α 1 + 2 + 3 =
Find the value of α for which the system has (i) a unique solution (ii) no solution (iii)
infinitely many solutions.
6 x 4x x 1 + 2 + α 3 =
3 2x x 2 x 1 − 2 + α 3 =
5 x 3x x α 1 + 2 + 3 =
Find the value of α for which the system has (i) a unique solution (ii) no solution (iii)
infinitely many solutions.
Linear Algebra
4. a) The Gauss elimination method is used to solve the system of equations
6 x 4x x 1 + 2 + α 3 =
3 2x x 2 x 1 − 2 + α 3 =
5 x 3x x α 1 + 2 + 3 =
Find the value of α for which the system has (i) a unique solution (ii) no solution (iii)
infinitely many solutions
6 x 4x x 1 + 2 + α 3 =
3 2x x 2 x 1 − 2 + α 3 =
5 x 3x x α 1 + 2 + 3 =
Find the value of α for which the system has (i) a unique solution (ii) no solution (iii)
infinitely many solutions
Linear Algebra
4. a) The Gauss elimination method is used to solve the system of equations
6 x 4x x 1 + 2 + α 3 =
3 2x x 2 x 1 − 2 + α 3 =
5 x 3x x α 1 + 2 + 3 =
Find the value of α for which the system has (i) a unique solution (ii) no solution (iii)
infinitely many solutions.
6 x 4x x 1 + 2 + α 3 =
3 2x x 2 x 1 − 2 + α 3 =
5 x 3x x α 1 + 2 + 3 =
Find the value of α for which the system has (i) a unique solution (ii) no solution (iii)
infinitely many solutions.
Linear Algebra
let u= (2,3,-1) and v= (0,-2,3)
a. determine u+v=
b. -2u=
a. determine u+v=
b. -2u=
