Final exam – practice problems
Linear independence
1. Check if the set of vectors {α1, . . . , αn}is linearly independent if
•α1= [1,2,3], α2= [3,6,7];
•α1= [4,−2,6], α2= [6,−3,9];
•α1= [2,−3,1], α2= [3,−1,5], α3= [1,−4,3];
•α1= [2,−3,1], α2= [3,−1,5], α3= [1,−4,3], α4= [1,1,3];
2. Check if the polynomials 1, X, X2, . . . , Xnare linearly independent.
3. Check if the polynomials α1= 2X+ 1, α2=X2, α3= (X+ 1)2are linearly independent.
4. Check if the rational functions 1
X,1
X−1,1
X−2, . . . , 1
X−nare linearly independent.
5. Check if the functions f1(x) = 1, f2(x) = sin x,f3(x) = sin 2xare linearly independent.
6. Find values of afor which the vectors α1= [1,2,−1], α2= [0,1,3], α3= [1,1,0] are linearly independent.
7. Supose that the vectors α1, . . . , αnare linearly independent. Check if the vectors β1, . . . , βnare also linearly independent,
if
•β1=α1, β2=α1+α2, β3=α1+α2+α3, . . . , βn=α1+α2+. . . +αn;
•β1=α1+α2, β2=α2+α3, . . . , βn−1=αn−1+αn, βn=αn+α1.
Systems of linear equations
1. Solve the following systems of equations.
x+ 4y+ 10z+ 20t=x
−6y−20z−45t=y
4y+ 15z+ 36t=z
−y−4z−10t=t
,
x+ 4y+ 10z+ 20t=−x
−6y−20z−45t=−y
4y+ 15z+ 36t=−z
−y−4z−10t=−t
,
2. Find bases for subspaces of solutions of the following systems of equations:
(X1+X2−X3= 0
2X2−X4= 0 ;(X1+X2−X3= 0
2X2−X4= 0 .
3. Depending on afind dimensions of subspaces of solutions of the following systems of equations:
2X1+X2+X3= 0
X1+ 2X2+X3= 0
X1+X2+aX3= 0
,
X1+ 2X2+ 3X3+ 4X4= 0
4X1+aX2+ 3X3+ 4X4= 0
2X1+X2+ 2X3+X4= 0
,
4. Solve the following systems of equations:
2x−3y+ 5z+ 7t= 1
4x−6y+ 2z+ 3t= 2
2x−3y−11z−15t= 1
;
2x+ 5y−8z= 8
4x+ 3y−9z= 9
2x+ 3y−5z= 7
x+ 8y−7z= 12
;
3x+ 4y+z+ 2t= 3
6x+ 8y+ 2z+ 5t= 7
9x+ 12y+ 3z+ 10t= 13
;
3x−5y+ 2z+ 4t= 2
7x−4y+z+ 3t= 5
5x+ 7y−4z−6t= 3
;
3x−2y+ 5z+ 4t= 2
6x−4y+ 4z+ 3t= 3
9x−6y+ 3z+ 2t= 4
;
8x+ 6y+ 5z+ 2t= 21
3x+ 3y+ 2z+t= 10
4x+ 2y+ 3z+t= 8
3x+ 5y+z+t= 15
7x+ 4y+ 5z+ 2t= 18
;
x+y+ 3z−2t+ 3w= 1
2x+ 2y+ 4z−t+ 3w= 2
3x+ 3y+ 5z−2t+ 3w= 1
2x+ 2y+ 8z−3t+ 9w= 2
;
2x−y+z+ 2t+ 3w= 2
6x−3y+ 2z+ 4t+ 5w= 3
6x−3y+ 2z+ 8t+ 13w= 9
4x−2y+z+t+ 2w= 1
;
6x+ 4y+ 5z+ 2t+ 3w= 1
3x+ 2y+ 4z+t+ 2w= 3
3x+ 2y−2z+t=−7
9x+ 6y+z+ 3t+ 2w= 2
.
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