Linear Algebra Answers

Let

T:

R3→R2

be a linear transformation such that

(1,1,1) = (1,0),

(1,1,0) = (2, −1) and

(1,0,0) =

(4,3). What is

(2, −3,5)?

 In each of the following cases explain whether R²"\\to" R is a linear transformation, if it is, supply a proof, if not, supply a counter ample

a) T(a, b) =a + b

b) T(a, b) =ab

c) T(a, b) =|a|²

d) T(a, b) =a - b

Consider the linear eigenproblem Ax=λx for the matrix

D=[1 1 2

2 1 1

1 1 3 ]

1. Solve for the largest (in magnitude) eigenvalue of the matrix and the corresponding eigenvector by the power method with 𝑥(0)T=[1 0 0]

2. Solve for the smallest eigenvalue of the matrix and the corresponding eigenvector by the inverse power method using the matrix inverse. Use Gauss-Jordan elimination to find the matrix inverse.

Is a rank of a matrix can be zero and what is nullity of a matrix?

Let A =

(8 -1 2

2 0 -5)

B =

(-1 7

3 -2

1 5)

and C =

(2 1

3 5)

(a) Calculate AB and A + B if they exist.

(b) Verify that (AB)C = A(BC).

(c) Calculate C-1 A.

Consider the linear eigenproblem, 𝐴𝑥=𝜆𝑥, for the matrix

D=[1 1 2

2 1 1

1 1 3 ]

1. Solve for the largest eigenvalue by the direct method using the secant method. Let 𝜆^((0))=5 and 𝜆^((1))=4

2. Solve for the eigenvalues by the QR method

Consider the linear eigenproblem, 𝐴𝑥=𝜆𝑥, for the matrix

D=[1 1 2

2 1 1

1 1 3 ]

1. Solve for the largest eigenvalue by the direct method using the secant method. Let 𝜆^((0))=5 and 𝜆^((1))=4

2.Solve for the eigenvalues by the QR method

Show that the map T : R^

2 −→ R^2 defined by T(x1, x2) = (x1 −x2, 0) is a linear transformation, hence find the matrix

of this transformation relative to the basis B = {(−1, 1),(0, 1)}

Write v = (2, -5 , 3) as a linear combination of

u1=(1,−3 ,2)

u2=(2,−4 ,−1)

u3=(1,−3 , 7)

Write the equations in matrix form :

x-y=-5

3x+2y=-5