Linear Algebra Answers

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.

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.

1.Solve for X from the matrix equation below. Here l is the identity matrix and det(B) ≠ 0 and det(A) ≠ 0.

B(X - l)A + B = A

Choose the correct option:

1. No such matrix X.

2. X=A–¹ B - A.

3. X=A–¹ - B +A.

4. X=A–¹ + B -A.

5. X= -A–¹ + B–¹ + l.

6. X= -(A–¹ + B–¹).

2. Consider the following linear system:

2x - 3y = -1

2x - 3y = 1

1. x=0 and y= 0 satisfy the system.

2. The system has exactly one solution.

3. The system is inconsistent.

4. The system has infinitely many solution.