Linear Algebra h.w! I have the questions

Linear Algebra h.w! I have the questions, I need answers and explanation of how they got the answer

1. Evaluate the determinant of the following matrices




3 3 0 5
2 2 0 −2
4 1 −3 0
2 10 3 2





 −3 0 72 5 1
−1 0 −3




2. Use Cramer’s rule to solve the system

x – 4y + z = 6
4x – y + 2z = -1
2x + 2y – 3z = -20

3. Find a unite vector in the direction of V and in the opposite direction.

v = (3,−2, 4, 1)

4. Let u = (1, 2, 3) v = (−1, 4, 0) and w = (−2, 1, 5)

(a) (u ·v)w =
(b) u×v =
(c) u · (v ×w)

5. Find the distance between u = (9, 4, 1, 7, 3) and v = (−1, 4, 0,−3, 9)

6. Find the cosine of the angle between the vectors (6, 2, 5, 9) and (−3, 1, 4, 0)

7. Find a unit vector orthogonal to the vectors
v = (1,−1, 1, 2) u = (2,−1, 0, 1), w = (1, 3,−1, 0)

8. Let u = (−2, 6, 4) and a = (2,−1, 0). Find:

projau =

9. Find vector and parametric equation of the plane containing the point
u = (−1, 0, 1) and is parallel to the vectors v1 = (0, 1, 1) and v2 =
(1, 0, 1).

10. Find the equation of the plane that passes through the points (0, 0, 1),
(1, 0, 0) and (0, 1, 0).

11. Find the parametric equation of the plane that passes through the point
(0, 0, 1) and is parallel to the vectors (0, 1, 0) and (0, 0, 1).

12. Are the vectors u, v, w form orthogonal set ?
If you do not explain your answer you will receive 0 points.

u = (1, 1, 1, 1) , v = (1,−1, 1,−1) w = (3,−3, 0, 0)

13. Write equation to the plane that pass through the points
P (1, 0, 4), Q(−1, 4, 3), R(2, 6,−2).

14. Find the parametric equation of the line that passes through the point
(2, 1, 3) and is parallel to the vector (1, 0,−5)

15. Decide if the planes 3x−y +z = 4 and y−x−2z = 2 are perpendicular.

16. Find the area of the triangle with vertices (0, 0, 1), (1, 0, 0) and (0, 1, 0).

17. Decide if the the set of vectors are linearly independent.

(a)

v1 = {(1, 2, 3), v2 = (1,
√

2,
√

3), v3 = (1,
√

5, 3), v4 = (0,−1, 2)}

(b)
u1 = {(1, 2, 3), u2 = (4, 5, 6)}

(c)
{(w1 = 1, 0, 0), w2 = (0, 1, 1), w3 = (0,−1, 1)}

18. Determine if the following functions form linearly independent set

(a) S = {ex, e3x}
(b) S = {1, sin x, sin 2x}

(c) S = {1, cos 2x cos2 x}

19. Determine if the following set is a subspace:

(a) The set of pairs (x, y) where y ≥ 0
(b) The set of Polynomials P (x) = a0 + a1x + a2x

2 + a3x
3

i. for which a0 + a1 = 0

ii. a3 −a2 = 1
iii. P (3) = 0

iv. P (0) = 3

(c) The set of continuous functions f = f(x) on [−2, 3] such that∫ 3
−2

f(x)dx = 0

20. Determine if the vector is in the span.

(a) Is u = (6, 2,−12) is in span[(3, 5,−1), (0, 12, 15)]
(b) Is 2×2 + 3x− 1 is in span[x2 + 2x + 3,−x2 + x + 2]

21. Find the coordinates of w = (1, 1) in the
basis v1 = (2,−4) and v2 = (3, 8) write vector as a linear combination
of vectors is a set is a basis

22. Find the dimension of the solution set to the homogenous system equa-
tions

(a)
3x + y – z = 0
-2x – y + 2z = 0
-x + z = 0

(b)
-2x + 3y – z + 2w = 0
3x – 2y + 3z + 2w = 0

23. Find the matrix of the following transformations in R2 that reflects a
vector about the line y = x and then reflects that vector about the
x-axis.