13.E: Exercises

Exercise \(\PageIndex{1}\)

Determine whether the given sets of vectors are linearly dependent or linearly independent. If they are linearly dependent, write one as a linear combination of the others.

a.) \(\vec u = (0,2), \vec v = (0,3)\)

b.) \(\vec u = (1,2,4), \vec v = (1,-2,3), \vec w = (-2,0,1)\)

c.) \(\vec u = (7,5), \vec v = (1,2), \vec w = (3,-1)\)

d) \(\vec u = (1,2,3), \vec v = (2,4,6), \vec w = (4,1,-4)\)

Answer

a.) linearly dependent

b.) linearly independent

c.) linearly dependent

d.) linearly dependent

Exercise \(\PageIndex{2}\)

Are the following sets subspaces of \(\mathbb{R}^3\) or not? If not, give an explanation and if so, prove it. 

a.) \(V\) is the set of all \( (x,y,z)\) such that \(x=0\)

b.) \(V\) is the set of all \( (x,y,z)\) such that \(2x=3y\)

c.) \(V\) is the set of all \( (x,y,z)\) such that \(x=6\)

d.)  \(V\) is the set of all \( (x,y,z)\) such that \(x+y=0\)

e.)  \(V\) is the set of all \( (x,y,z)\) such that \(x+y=2\)

Answer

a.) yes

b.) yes

c.) no

d.) yes

e.) no

Exercise \(\PageIndex{3}\)

Is \(\vec v = (1,0,-1)\) in the span of \(\{ (5,3,4), (3,2,5)\}\)? If so, write \(\vec v\) as a linear combination of the two vectors. 

Exercise \(\PageIndex{4}\)

Prove that if a finite set of vectors contains the zero vector, then that set is linearly dependent.

Answer

Hint: To prove a set of vectors is linearly dependent, you need to show that one of the vectors is a linear combination of the others. 

Exercise \(\PageIndex{5}\)

Do the following sets of vectors form a basis for \(\mathbb{R}^n\)? Why or why not?

a.) \(\vec v_1 = (4,7), \vec v_2 = (5,6)\)

b.) \(\vec v_1 = (1,-3), \vec v_2 = (-3,9)\)

c.) \(\vec v_1 = (4,7,4), \vec v_2 = (5,6,0), \vec v_3 = (2,-1,1)\)

d.) \(\vec v_1 = (4,-1,4), \vec v_2 = (5,2,0)\)

e.) \(\vec v_1 = (4,7,4), \vec v_2 = (5,6,0), \vec v_3 = (2,-1,1), \vec v_4 = (0,1,2)\)

Answer