Linear Algebra Answers

Which of the following linear systems is homogeneous. [just give the answer I) or II) or III) : no motivation

is required.]

I) 3x + 14y = 0 & 2x − 3y = 7

II) 7x + 2y = 10 & 11x − 0y = 7

III) 4x − 23y = 0 & 7x − 3y = 0

Let X =

1 2

3 4

; E =

a

b



Find each of the following.

If the operation cannot be done : state undefined operation.

a) XE

b) EX

c) XT X where XT

stands for the transpose of X

Consider the linear equation 2a + 3b = 4

Is (a; b) = (1; 1

2

) a solution to the equation? Motivate your answer.

If we have a linear equation what is the power of the variable(s) involved

True or false :3Z= Z+Z+Z when Z is the matrix

Let 0 Zero be the zero matrix. Write down the matrkx if of size 2×2

Show that the set {1 − 𝑥, 3 − 𝑥 2 , 𝑥} spans 𝑃2

Determine the basis for the set of vectors lying on the plane 𝑊 = {(𝑥, 𝑦, 𝑧): 3𝑥 + 𝑦 − 5𝑧 = 0}

Find the nullity and the range of the space spanned by 𝑢 = (1,0,2), 𝑣 = (−3,1,1), 𝑤 = (0,1,4) and 𝑥 = (−1, −3,5)

Find the dimension and a basis for 𝑊 = {(𝑥, 𝑦, 𝑧, 𝑤,𝑡): 𝑥 + 𝑦 + 𝑧 + 𝑤 + 𝑡 = 0, 𝑥 − 𝑦 + 𝑧 − 𝑤 + 𝑡 = 0}