Linear Algebra Answers

find the sum and product of eigenvalues of the matrix

[1 2 3

-1 2 1

1 1 1 ]

Let φ : V → W be a linear transformation of vector spaces over the field F. The kernel of φ is by definition

the set ker(φ) ⊂ V of vectors v in V such that φ(v) = 0. The image of φ is the subset im(φ) of vectors w ∈ W

for which there exists some v ∈ V such that φ(v) = w.

(1) Show that the kernel of φ is a subspace of V .

(2) Show that the image of φ is a subspace of W.

(3) Show that φ is injective if and only if the kernel is 0.

(4) Show that φ is surjective if and only if the image is W.

Solve the given linear system by any method. 3x1 + X2 + X3 + X4 = 0, 5x1 - X2 + X3 - X4 = 0

Show that T(x1, x2, x3, x4) = 3x1 −7x2 + 5x4 is a linear transformation by finding the

matrix for the transformation. Then find a basis for the null space of the transforma￾tion.

determine whether the homogeneous system has nontrivial solutions by inspection (without pencil and paper).

 2x1 − 3x2 + 4x3 − x4 = 0,

7x1 + x2 − 8x3 + 9x4 = 0,

2x1 + 8x2 + x3 − x4 = 0

solve the linear system by gauss-jordan elimination

 − 2b + 3c = 1 ,

3a + 6b − 3c = −2,

6a + 6b + 3c = 5

solve the linear system by gaussian elimination

 x − y + 2z − w = −1,

2x + y − 2z − 2w = −2 ,

−x + 2y − 4z + w = 1 ,

3x − 3w = −3

5. Let φ : V → W be a linear transformation of vector spaces over the field F. The

kernel of φ is by definition the set ker(φ) ⊂ V of vectors v in V such that φ(v) = 0.

The image of φ is the subset im(φ) of vectors w ∈ W for which there exists some

v ∈ V such that φ(v) = w.

(a) Show that the kernel of is a subspace of V .

(b) Show that the image of is a subspace of W.

(c) Show that is injective if and only if the kernel is 0.

Let {u1, u2, ..., un} be an orthogonal basis for a subspace W of R

n

, and let T : R

n → R

n

be defined by T(x) = projW (x). Show that T is a linear transformation.