Problems:
Vectors, matrices, and polynomials
1) Use cofactor expansion to find the determinant
| 4 -1 1 6 |
| 0 0 -3 3 |
| 4 1 0 14|
| 4 1 3 2 |
2)Find the characteristic equation, eigenvalues, and corresponding eigenvectors for
A = 1 2
2 1
3) Answer True or False for the following statements: (True means always true; False means sometimes false). Justify your answers. Remember, you can use counterexamples only to justify False)
a) Statement: "If T: R^n -> R^n is a one-to-one linear operator, then T is invertible".
b) Let Ta: R^n -> R^n be a linear operator with standard matrix [Ta] = A.
Statement: "If det (A) = 0 then the range of Ta = R^n. ".
c) Consider the following linear operations on R²:
T1 is a counterclockwise rotation of 60º, and T2 is an orthogonal projection on the y-axis is R².
Statement: "T1ºT2 = T2ºT1"
d) Statement: "T(x, y, z) = (2x, 0, 1) is a linear operator in R³."
4) Answer True or False for the following statements: (True means always true; False means sometimes false). Justify your answers. Remember, you can use counterexamples only to justify False)
a) The set of all invertible 3x3 matrices is a subspace in the vector space of all 3x3 matrices under standard operations of matrix addition and scalar multiplication.
b) The set of all polynomials a0 + a1x + a2x² + a3x³ for which a0= 0 is a subspace in the vector space R3[X] of all polynomials of degree ≤3 under standard operations of polynomial addition and scalar multiplication.
c) The set of all polynomials a0 + a1x + a2x^2 + a3x^3 for which a3 ≠ 0 is a subspace in the vector space R3[X] of all polynomials of degree ≤3 under standard operations of polynomial addition and scalar multiplication
5)Determine whether w is in span of the given vectors v1, v2,.. , and if it is, express w as a linear combination of v1, v2, ...
a) w = {-1, 4, 15}, v1 = ( 1,2,8) v2 = (3,0,1)
b) w= 4x+6x², v1 = 1 + x + x², v2 = 2 + 2x + 3x³, v3 = 1+5x+6x²
c) w= 0 5 v1= 1 2 v2= 3 1
-4 -1 -1 0 1 1
6) Determine whether the given vectors span R³.
a) (1,3,4), (2,0,1), (2,1,0)
b) (1,5,1), (2,6,1), (3,3,0), (4,6,2)